István Gulyás
THE DOCTRINE
OF THE MODERN BOOKKEEPING
Elements of theory
of the modern n-entry (n³3) special and the
general bookkeepings
(Death of the account theories)
2010
Publisher: István Gulyás; 1163-H. Budapest, Edit St. 15.
Author and translator: István Gulyás economist.
http://www.ginprofessional.hu ; mailto:gulyas@ginprofessional.hu
Budapest, 2010, second revision edition; published at the author own expense, firstly in 2009; in Hungary.
István GulyásThe axiomatic system of the N-fold (N>=3) bookkeeping by scientific work is licensed under a Creative Commons Name it!-Do not sell it!-Do not change it! 2.0 UK: England and Wales License.
Based on a work at www.ginprofessional.hu.
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István Gulyás
Economist
The author in 2009
(Born: 17 October 1948)
The original title of this work in Hungarian language:
Gulyás István
MODERN KÖNYVVITELTAN
A modern n-szeres (n³3) vagyonkönyvvitel,
mint
az egyik speciális könyvvitel
elméletének elemei
és
axiomatikus rendszere
(a számlaelméletek halála)
ISBN 978-963-88486-6-6 (book) - 08/17/2010
ISBN 978-963-88486-7-3 (online)
ISBN 978-963-88486-8-0 (CD)
"The things may show themselves surprisingly to other from new
viewpoint such as they were ever when we met them. This one
holds on the bookkeeping too."
István Gulyás
CONTENTS
PREFACE OF THE EDITION 2
0) (T10/C). 
=
=...=
≥0, that is: if we classify to the interval (0,t] belonging property changes base class and/or to the tth time-point (t=1,2,...) belonging balance class of it, n ways (n³
2), namely by arbitrary but differing property aspect A1,A2,..,An, or if we complete the property classification system with a classification by aspect An+1, then the structures of classifications of this property classification system are differing, while the main sums of it, which ones expressed by same measure, are all equal (T16). 
Ai=CE+CF≥0, that is: if we classify set of balances (that is objects of the property) of to the interval (0,t] belonging gross property changes by two differing, that is: by assets and capital aspect it is classified, then this two structures of classifications system are also different, but the in same measure expressed two main sums are equal. (T16/C1).
Corollary 2: 
I(t)=
Ai=CE+CF≥0, that is: if set of to the interval (0,t] belonging gross property changes by time, when the set of to the tth time-point belonging balances (that is objects of the property) by assets and capital, that is together: it is classified by three differing aspect, then the structures of classifications of this dynamic and static property classification system are different, but the in same measure expressed three main sums are equal. (T16/C2).
Corollary 3: TM=AM=CM=...=XM≥0, that is: if the set of to the interval (0,M] belonging gross property changes by time, when the set of to the Mth time-point belonging balances (that is objects of the property) by assets and capital plus other aspect, that is together: it is classified by N differing (N³
3 and integer) aspect, then the structures of classifications of this dynamic and static property classification system are different, but the in same measure expressed main sums all are equal. (T16/C3). 
Ai(t)=
CE(t)+CF(t)≥0, that is, to the tth time-points (t=1,2,...,M) belonging the same unit and A-C-aspect main sums of TA-TC-aspect dynamic property classification system of the gross property and their to Mth time-point summarized sums are equal (T17). I(t)=Ji(t)=VS(t)+VI(t)≥0, that is, to the tth time-points (t=1,2,...,M) belonging the same unit and A-C-aspect main sums of T-TA-TC-aspect dynamic property classification system of the gross property and their to Mth time-point summarized sums are equal (T18). Ai(t)=CE(t)+D(t)³
0 (where PGR(M)=Ai(t)D(t)³
0 and CE(t)0; and t=1,2,...,M; i=1,2,...,n) formula represents the law of the material position. Mean of this law: The man and all other economist, from its birth to its death, its existence in all tth moment, (1) has gross property [PGR(M)>0], bat then has debt also [D(t)>0], (2) D(t) in the good case is significantly less, in bad case is greater then the gross property, (3) or it does have neither its property [PGR(M)=0] its debt [D(t)=0] (this time it is pauper), (4) or its position worse at this one also because it has only debt [PGR(t)=0, D(t)>0] (this time it is pauper debtor). (5) And other case is not possible. (6) The material position of the economist and its all factors change in the time, the economist either economizes or leaves to itself its property, hence (7) its property, as its material position's either main factor, may investigate by N aspect (N≥3) , that is, at least by time, asset and capital-aspect (T29). 
3.1 Principles
THIRD PART
My book[1], with exception of its third part, was finished by the end of 2003, naturally Hungarian.[2] The
scientific problem and the foreknowledge of the result already born in 1997; but I begin to write the book only in 2000.The content, at that time, was the first and second part just as the appendix of the book, which was written on purpose to let it popularize this doctrine but let it not claim more knowledge then the popular education.
At that time seemed, I explicated already all important rudiments what I could tell with the traditional bookkeeping (aka: accountancy) or rather the modern N-entry (N-fold) (N≥3) property bookkeeping related.
This time I read firstly the book[3] of Gábor Szász whose title is 'The axiomatic method' and which in 1972 issued.
This section is about to how to became a true science of mathematics. Here Szász explicate that: The Egyptians and the Babylonians have not the rules on their mathematical knowledge. That is, speaking of today's language, they are not established theorems only edited concrete numerical model examples, and on ones demonstrated the methods of calculation. The expressing forms of the today's mathematics, which ones in the schools are used as general expressions, for example the definition or the theorem or the axiom and the proof, developed out yet in the Ancient Greek culture. Meanwhile, the mathematics instead of empirical collection of knowledge has become a real deductive science.I realized promptly that the system of concepts of doctrine of the traditional bookkeeping is not exact. The bookkeeping's traditional doctrine, neither before Paccioli[4] nor from Paccioli ere now, has not nicely concepts and from this ones standing and uncontradicted and the system of built on each other concepts. This one has not axioms and verifying and built on each other theorems which ones form a coherent system. But this one is true on the so far completed modern doctrine of the N-entry (N≥3) property bookkeeping too,
although it has exact definitions, axioms, theorems with them certain relations, but yet on the all theorems extending without proofs.Thus the bookkeeping's doctrine, as a science, in the current state, as it is, does not exceed from the empirical collection of knowledge standing, before 2500 years Egyptian and Babylonian level of the mathematics.
But the possibilities were given from Euclid already at least to 2300 years. The possibility existed that let them describe on modern method the bookkeeping's doctrine also
like to the already then developed mathematics and geometry.However, the modern N-entry (N≥3) bookkeeping was discoverable in the antiquity also, but in time of Paccioli already on all cases. Namely any economic event also occurred in, already in the antiquity also, never not showed only two, that is, by assets and capitals aspects of changes of the property and/or debt. The number of showed aspects is always at least three. For example: if we bought some product on credit then from the dates of the economic event known promptly the values of three parameters: i.e. (1) the time-point of the change and (2) the bought asset's type and (3) the source of purchase of the asset. (That is that in the case (3): What is the invested capital? Is it the foreign capital from credit or equity?) The triple (as triple of the event coordinates) of these three parameters promptly and on naturally method denoted out and such created or changed by the time- assets- and capital-aspect natural property classes, to which ones touched by this economic event as by property change. These coordinates of the economic event defined on naturally method from the time-assets-capital-aspect property classifications standing complete just as dynamic and static property classification system. Let us name by this example classification system to three-pen balance sheet. The bookkeepers (aka: accountants) and the professors of the bookkeeping did not realize up these ones by more then 2000 years.
In turn the triples of coordinates are attributes of the economic events, ever since the man economy; and these were and are always by the bookkeeper (accountant) known into the set of dates of the bookkeeping, if they on clay-table booked too. Only they did not realize up this one nor.
The traditional bookkeeping and its doctrine more then 2300 years across did not develop sufficiently. Paccioli wrote[5] down firstly, in 1494, the use of a rudimentary double-entry bookkeeping. He showed in the use of this on simple example across. After this, Shär created, in 1890, a closed system of the accounts (in German: 'Das geschlossene Kontensystem"[6]). Schmalenbach[7] and Kosiol, in 1933, dreamed the so-called dynamic balance sheets, but these were in truth static balance sheets. Meanwhile doctrine of the bookkeeping, from Paccioli to Schmalenbach, got to the so-called account theories, which ones contain two or four sets of the accounts. [An example for the base equation of two sets of accounts is the follows: Assets=Capitals. And an example for the base equation of four sets of accounts is the follows: Assets=Liabilities+Equity+(Yields-Costs), where the each one term of the equation symbolize a set of accounts.] So the double-entry bookkeeping, over 400 years, altogether so much developed.
In fact, the traditional doctrine of the bookkeeping, from 1910 to present, that is a whole century across, such materialized respect to the said minor changes. Otherwise it, without significant evolution, stagnated. The scientists to present describe that the so-called single-entry bookkeeping holds and it may use as a complete bookkeeping, which is fundamentally error. I verify this one also in this work. And they were not able to break with account theories after appearance of the personal computers, on end of the twentieth century began in the age of PC nor. Moreover, the software developers simply only imitate with their accounting programs the manual double-entry bookkeeping, so they only conserved in the old bookkeeping's existing knowledge and practice. I proof here this one too.The evolution of the traditional bookkeeping and its doctrine, to present, got in dead end. What is more, this doctrine became straight orthodox. Therefore, early in 2004, I saw to unavoidable to attempt the setting up of elements of the bookkeeping. I saw that unavoidable to prove by the elements of the bookkeeping, that is, to prove by the bookkeeping's axiomatic system the existence of the N-entry (N≥3) property bookkeeping just as the its features and by it opened wide space of opportunities. I also demonstrate here, on exact mode, that both the single-entry bookkeeping and the double-entry bookkeeping are incomplete. This one mainly today, in the age of PCs, hampers the supply with sufficient information the economic actors.
I decided on base of these. I suspended the preparations publishing of my book and commenced the compilation of the accounting elements.
This has happened built on each other definitions, axioms, built on each other items with formulation and proof. Between these are more new too. This activity, which incorporates in coherent system the bookkeeping's elements, to a pleasant surprise, brought additional knowledge too. Although, in meantime illness and surgery and lengthy convalescence and my job prevented me in that I realize my goals. Now, however, I may publish the result here finally. This can be polished. This work can be refined or improved and expanded as well. (This second edition also shows this one.) But otherwise also it can be incorporated this axiomatic system. It is now already scientific commonplace and found fact. This is a found fact, if one compares for example the Euclidean and Bolyai-Lobachevski and Hilbert geometry, as axiomatic systems. However, it is also fact that the by me described axiomatic system of doctrine of modern bookkeeping will no longer be avoided and can not be ignored, in my judgement, neither in the education nor in the scientific research. So, with this moment the accounting's doctrine also crossed with the exact, clear and coherent concepts and axioms grounded between the deductive sciences. If 2300 years late, but over! The bookkeeping's this modern doctrine with this has joined to the advanced and exact so-called justificative sciences. Already it was time.It is important to note that: This theory system does not do unusable and unavailable the present bookkeeping praxis. However, it opens the way out of the development of this science before the modern satisfaction of the information's need of the management just as before the substantive development of the bookkeeping programs.
This work may be read my book (438 pages) in third chapter (the last ca. 100 pages); it one concentrated but high school level with knowledge also achieved doctrine of the modern accounting. Who wants a propagative-featured detailed explanations containing description I suggest to it read the first two parts of my book. Although as I mentioned this is currently accessible in Hungarian only.
I offer my book and its this part near the spontaneously interested readers to the
***
And finally let us look the following short story:
Nobody can not be prophet in the own country! Consequently I also can not be.
I offered my book on Hungarian and foreign edition (see in the 5th appendix the 2nd e-mail) to the Akadémia Kiadó Zrt. (to the Academy Publisher Co.) because I had no enough money to the private publication and on the requisitioning of a translator. The publisher firstly (on Thursday, September 03, 2009) favorably accepted the offer sithence this time requested authorization on previous examination of the handwriting by its economist experts (see in the 5th appendix the 3rd e-mail) with approximately one month period. I gave the authorization yet on the same day (see in the 5th appendix the 4th e-mail). I recommended yet on the next Tuesday that let the publisher apply, if possible, a mathematician expert too (see in the 5th appendix the 5th e-mail). They thanked this suggestion within a half-hour (see in the 5th appendix the 6th e-mail). But to my surprise, yet on the same day three hours later, they sent an e-mail message that they do not intend to publish my book by the judgment of the experts (see in the 5th appendix the 7th e-mail). Justification was not.
I suspect that the some director of the publisher a mathematician expert on application done my offer considered to brusqueness or that writer, whose I nibbled at her work, swung the case.
Let us remember the case of János Bolyai. He wrote his work from the non-Euclidean geometry in Latin language by then practice. This appeared in the appendix of book of his father to which title is Appendix of Tentamen. The work of János Bolyai unread lay on the shelf of the Hungarian Scientific Academy by decades without Hungarian translating. Finally, on great shame of president of the academy, a foreign publisher requested authorization to translate on French, English and Italian languages the famous mathematical work which yet then neither was translated on Hungarian language.
But what do I want? I am not János Bolyai and my book does not speak on the absolute science of the space. Still I so thought that the scientific world and the student of the theme must know the modern doctrine of the N-entry bookkeeping and its axiomatic system. Hence I started the translating of this part of my book on English language, firstly in my life. I know that this translating is not unfaulty; hence I apologize to the reader. But failing enough money I can realize my target only thus.
However, I hope that this work raises the interest of the sponsors and/or the publishers.
Budapest, 23 August 2010
István Gulyás
Economist
The grounds of the modern bookkeeping doctrine:
The n-entry (n
its axiomatic system
1. The elements of the property theory of the bookkeeping
Principles
1.111 The concepts of theory of the general bookkeeping
1.112 The concepts of property theory of the special property bookkeeping
1.121 The axioms of the property and other chronologic sets
P.[38]: 1./T1.
P.: 1./T
P.: 1./T1.
P.: 1./T1, T
11, T16, T19, T28.P.: 1./T
11, T16, T27, T28.P.: 1./ T
1, T16, T17, T18.P.: 1./T
15, T17, T29.1.122 Debt axioms
P.: 1./T
P.: 1./T
2, 3./T1, T3, T4.P.: 1./T
2.P.: 1./T
15, T17.1.123 Economic and general event axioms
P.: 1./T
P.: 1./T
15, T19.P.: 1./T
19, 2./T7.P.: 1./T
9, T19. 3./T6.P.: 1./T
17, T22.property theory and their proofs
The proofs of the following property theory theorems have with such important feature that, in these, I do not refer to bookkeeping rules, although in these ones, I tell from the property theory of the bookkeeping. Namely, this one is my basic strategical target when I build up the bookkeeping's property theory, because only such it may be to show out convincingly that the main factors of the material position, that is: the property and the debt and their character of the changes in the time, define the nature of bookkeeping and not inversely.
Attribute classifications and their features of the classes
Theorem 1: If the economist has property in the tth time point (t=1,2,...), then in this property or rather in its any non-empty assets-aspect static property class belonging goods' quantity or monetary value (or other characteristic's measure) marker main or part sum is possible only positive number[41] (T1).
By the condition, just as according to the inventory and/or the books the economist has property[42] in the tth time point (t=1,2,...). Let us class, fully, in the property belonging goods by assets-aspect (other by property types) i (i=1,2,...) different (see: axiom A6) to static final property classes. Let
C be non-empty and undivided set (base class) of goods of this property. Additionally, let C1,C2,...,Ci,...,Cn be the all pairwise disjoint final classes of C, which ones symbolize the non-empty final classes of types of the objects of property.Now let us order to the single final property classes, separately, in the tth time point into the property classes categorized property's objects' quantities, which ones are denoted by qi (for all i), and monetary values, which ones are denoted by vi (for all i too), as this property classification's on classes defined values of functions. Let qi(Ci) denote so the quantity of the objects of property, as function of Ci, by types of property (that is, by final classes). Let it express the total sum of these quantities qi(Ci) (for all i), aside from denoting of single natural measures and the tth time point by follow equality:
q1(C1)+q2(
C2)+...+qi(Ci)+...+qn(Cn)=
qi(Ci).
Now let 
(C) denote this classification's the main sum, which expresses the total quantity of goods, as function of C.
Additionally let vi(Ci) (for all i) denote value of quantity of the types of goods, which ones expressed in same currency. This time let it expresses the total sum of vi(Ci) for all i:
v1(C1)+v2(C2)+...+vi(Ci)+...+vn(Cn)=vi(Ci).
Let 
(C) denote this classification's yon main sum, which expresses the total monetary value of goods, also as function of C.
Let p1(C1),p2(C2),...,pi(Ci),...,pn(Cn) denote still the (average) unit prices of single types of the goods. The pi(Ci) (for all i) is function of Ci, where pi denotes the (average) unit price.
Now let us show that
(C)=q1(C1)+q2(C2)+...+qi(Ci)+...+qn(Cn)=qi(Ci)>0,(C)=v1(C1)+v2(C2)+...+vi(Ci)+...+vn(Cn)=
vi(Ci)>0. In the non-empty assets-aspect final property class Ci (for all i) and in the tth time point existing property's objects' quantity expressing part sum qi(Ci) (for all i) may be only positive number, by axiom A1. But the unit price pi(Ci) which belongs the final property class Ci (for all i) also may be only positive number, by axiom A3.
Thus clear that
qi(Ci)∙pi(Ci)=vi(Ci)>0 for all i, because the product of positive numbers is positive.
Consequently: qi(Ci),pi(Ci),qi(Ci)∙pi(Ci)=vi(Ci)>0[43] (for all i), just as since the sum of the positive numbers is positive, so qi(Ci)>0 and vi(Ci)>0 (for all i).
Additionally: The next statement applies by axiom A4:
"If on classes of a property classification, which ones are pairwise disjointed, is interpreted some measure function (or its positive coefficient linear transformation), then by this function to the final classes ordered sum of part sums equal to the base class ordered with main sum (A4)."
This time the follow two equalities are true:
(C)=q1(C1)+q2(C2)+...+qi(Ci)+...+qn(Cn)=qi(Ci), just as
(C)=v1(C1)+v2(C2)+...+vi(Ci)+...+vn(Cn)=qi(Ci)∙pi(Ci)=vi(Ci).
Now since (C)=qi(Ci) and qi(Ci)>0 is true, just as because (C)=vi(Oi) and vi(Oi)>0 is also true, consequently (C)>0 and (C)>0 is also true.
But we obtain also this, if we sum the qi(Ci)>0 numbers for all i and the vi(Ci)>0 numbers for all i, since the sum of the positive numbers is positive.
Consequently the (1), (2), (3) and (4) statements and thus the theorem 1 are true. Q.e.d.
P.[44]: 1./T2, T3, T4, T5, T6, T7, T8, T9, T11, T12, T14, T15, T16,
T18, T19, T21, T22, T23, T24, T29, 3./T1
C.[45]: 1./A1, A3, A4, A6.
Remark: In the future, I will express briefly and together the main sum and part sums of the assets-aspect property classification on the follow method:
For the formula (C)=qi(Ci) and for the formula (C)=vi(Oi) I will write only the follow formula: P=Ai, where Ai denotes (i=1,2,...,n) in the quantity or the monetary value (etc.) some part sum of asset type class of the assets-aspect property classification; and P=(C) and P=(C) also together, if the previous notations are unnecessary.
Theorem 2: If the economist has debt (other: liability or foreign capital) in the tth time point (t=1,2,...), then in its gross property's capital-aspect static middle class, whose name is still foreign capital class, or rather in its any non-empty middle or final class belonging part's quantity or monetary value (or other characteristic's measure) marker main or part sum is possible only positive number[46] (T2).[47]
Let E0 denote the economist who has debt in tth time point by the condition and e.g. by the inventory. Let D0 denote debt's measure of E0 (that is the main sum of the static debt/liability class), aside from denoting of the tth time point.
Let us show that D0>0.
Inasmuch as the economist E0 has debt, hence necessarily it has creditor by axiom A9. Let this creditor be e.g. now the economist E1 by oneself, to who the economist E0 owes with D0. Since E1 is creditor of E0, hence E1 has acknowledged claim CL1 opposite the E0. Its claim CL1 is same measure and equal with D0 (CL1=D0), by axiom A10. Namely the axiom A10 says that: "In a given time existing debt of the debtor equal its creditor's or creditors' opposite with it with now existing total claim (A10)". So a part or the whole of gross property of E1 materializes as claim CL1 in opposite E0 (by A10). That is, the follow statement is true: CL1≤P1, where P1 denotes the gross property of E1 (see: figure T2).
|
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D0 |
= |
CL1 |
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D0 |
P1 |
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E0 |
E1 |
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Figure T2 |
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However we know in the theorem T1 that: "If the economist has property in the tth time point (t=1,2,...), then in this property or rather in its any non-empty assets-aspect static property class belonging goods' quantity or monetary value (or other characteristic's measure) marker main or part sum is possible only positive number." Consequently the existing gross property P1 of E1 greater then zero, that is P1>0, and this respects to any part sum of assets-aspect non-empty property class of this gross property P1, thus this is true to claim CL1 of E1 too. That is, CL1>0 is also true. Since CL1>0 and CL1=D0 hence from these D0>0 flows directly.
By the same train of thought we can know to show that: the positivity exists to the all non-empty final classes of debt D0 belonging the part sums of types of debt too. Hence we may have: D0,1+..+D0,j+..+D0,N=D0>0, where D0,j>0 is one of the part sums of the non-empty static final debt/liability (other foreign capital) class.
Q.e.d.
P.: 1./T
3, T4, T5, T11, T12, T13, T14, T15, T21, T29.C.: 1./A
9, A10, T1.Theorem 3 (Lemma[48]): In the tth time point (t=1,2,....) the difference of the size of property of the economist and the in with it same measure expressed size of debt of the economist may be greater or less then zero, or equal to zero (T3).
Let P denote the size of gross property of the economist, as the main sum of assets-aspect static classification of its gross property, in some tth time point. P≥0 is true by T1 and A2. Item, let D denote the size of debt (aka: liability) of the economist, as the main sum of relative base class of capital-aspect static classification of its gross property, also in the tth time point. D≥0 is true by T2 and A2. Additionally: the measures of P and D are same.
Let us show that P-D0 in the tth time point. (We set aside from t in the notations henceforth.)
It is true that P>D or P=D or P<D by axiom A8. Thus we can denote these together by the follow formula: PD. Now if we subtract from both side of inequality PD the debt D, then from this the truth of the formula P-D0 and thus the truth of the 3rd theorem flow directly.
Q.e.d.
P.: 1./T4, T29.
C.: 1./A2, A8, T1, T2.
Let PNE denote the size of net property of the economist in the tth time point. (E denotes the economist.) Item, let P denote the E's gross property's the size (P³ 0 by T
1 and A2), and let D denote E's debt's size (D³ 0 by T2 and A2), and these are defined also in the tth time point. The measures of these three variables PNE, P and D are same.Let us show that PNE
0 in the tth time point. (Henceforth we set aside from t in the notations.)
We know that P-D
0 by the theorem T3. But since the difference P-D is called to net property, by definition, and here it is denoted with PNE, hence P-D=PNE. However since P-D0 and P-D=PNE thus PNE0 is true. Q.e.d.
Remark: If in some tth time point PNE
=0, then to this main sum PNE belonging equity (other eigen capital) class CE is empty, by A2. But if PNE>0, then CE, which the capital-aspect middle class of the property, has element(s). This/these property object(s) is/are into CE, in the tth time point. If now PNE<0, then CE has element(s) also, but this/these is/are the missing property object(s) in the tth time point; this time just hence negative PNE, by the relating definition.P.: 1./T5.
C.: 1./A2, T1, T2, T3
Let PGR, CE, CF be three variable (CE=PNE) in any tth time point (t=1,2,...). Let PGR denote, as main sum, the size of the gross property in time point t. Let CE denote, as main sum, the size of the equity (other the eigen capital). Item, let CF denote, as main sum, the size of the debt/liability (other the foreign capital) also in tth time point.
PGR and CF is both non-negative by T1, T2 and A2. We may write that
(1) PGR-CF=CE, then after redistribution that
(2) PGR=CE+CF.
Firstly let us show that CE+CF≥0 in any tth time point.
Since PGR≥0 in any tth time point (by T1, A2), hence we may write that:
(3) PGR=CE+CF≥0 in any t.
Secondly let us show that CE0 and CF≥0, in CE+CF≥0 in any tth time point by the hypothesis and the (3).
Since CE0 and PGR-CF=CE, so we write:
(4) PGR-CF=CE0, that is:
(5) PGR-CF0 and from this: (6) PGRCF, in PGR≥0 and CF≥0 in any t.
We know from the axiom A8 that (a) PGR>CF or (b) PGR=CF or (c) PGR<CF and both non-negative. Consequently (4), (5) and (6) is true near (3), so (a) (b) and (c) is true too, because PGR and CF has not upper bound, that is, they may be whatever great numbers, only the zero exists as their lower bound. Hence e.g. if PNE=0, then CF=-CE (in CF≥0). This time the eigen capital (alias equity), in absolute value, same to the foreign capital (alias liability), but CE≤0.
Thus it is true that in any tth time point CE0 and CF≥0, in CE+CF≥0.
Q.e.d.
P.: 1./T11, T15, T29.
C.: 1./A2, A8, T1, T2, T4.
Theorem 6: In the tth time point (t=1,2,....) to the static final class of the net property which named startup capital belongs part sum may be only positive number (T6).
Let C
ST denote the part sum of startup capital class of the net property. Let us show that CST>0.C
ST shows by definition that when the economist began to manage, then how much was its invested gross property in money or in form of other asset, just as that, how much other property did it have to invest, as capital lifting, additional and eventual, or rather how much property did it have to extract, as capital cutting, from the economy.Now let the economist invest into its economy e.g. money and a realty on the beginning of the economy. Let PST denote the size of the assets-aspect startup property. C
ST=PST by definition and their measures are same. But PST>0 (by T1), thus we may write: CST=PST>0, so CST>0.Let C
ST' denote the value of the modified startup capital. In case of the capital lifting let D PST denote the size of the additional property and D PST>0 (by T1). Let D CST denote the plus capital. D CST=D PST by definition and their measures are same. Since D PST>0 and D CST=D PST>0 and thus D CST>0. The value of the lifted startup capital CST'=CST+D CST. But since CST>0 and D CST>0 is true, thus CST'>0 too.In case of the capital cutting let CST'=CST-D CST, where D CST is such capital-size which equal with that part of the property with which we extract among of CST and it is less then CST but greater then zero, that is 0<D PST=D CST<CST. Thus D CST-D CST<CST-D CST, hence 0<CST-D CST=CST' and thus 0<CST'.
That is true that CST,CST'>0.
Q.e.d.
P.: 1./T11, T12, T14.
C.: 1./T1.
Theorem 7: In the tth time point (t=1,2,....) to the static final class of the net property which named reserve capital belongs part sum may be only non-negative number (T7).
Let CR denote the part sum of reserve capital class of the net property. Let us show that CR>0.
CR shows by definition that the owner(s) or other(s) when and how much other property did it/they have to invest eventual into the economy, or how much property did it/they have to extract from there, not counting the startup capital.
Now let the economist invest into its economy e.g. money and/or other asset. Let PR denote the size of this assets-aspect additional (surplus) property. However CR=PR by definition and their measures are same. But PR>0 (by T1), so we may write: CR=PR>0, that is CR>0.
We may show alike that if reserve capital is reduced, then it is also greater then zero if there is still element in the reserve capital class, or else CR=0 (A2).
Q.e.d.
P.: 1./T7/C, T11, T12, T14.
C.: 1./A2, T1.
From the theorem it is clear: if CR>0 then part sums of all non-empty subclasses of reserve capital class are also positive numbers.P.:
C.: 1./T
7.Theorem 8: In the tth time point (t=1,2,....) to static cumulated gross result class of the net property, which named static cumulated yield class, belongs part sum, as in the tth time point existing quantity or monetary value (or its other characteristic's measure) of the cumulated yield, we may express with only positive number (T8).
Let YC denote in the tth time point (t=1,2,...) existing the size of the cumulated yield, which the economist realized in the interval (0;t]. Let us show that YC>0.
Under the cumulated yield (value) YC must be understand the increment of the eigen capital (alias equity) which was obtain in interval (0;t] (t=1,2,...) of economy, but not included the increment of the startup and/or reserve capital, by definition. This equity's increment may materialize in shape of any cash income or received goods or services (to barter also) or recognized claim, furthermore the property's natural increment or rather debit's release. This increment equal with the part sum YC of the cumulated yield class named to final class of the equity named to static relative base class in the tth time point.
Let this equity's increment materialize e.g. in shape of cash income and let IM denote this cash income's quantity. YC=IM by definition and their measures are same. We know (from T1) that the quantity or the monetary value of the property or its any part may be only positive number. Since this cash income is part of the property, so we may write: YC=IM>0, that is YC>0.
We may show alike that if the cumulated yield materializes e.g. in other form of the property, in the aforesaid, then YC>0 also.
Q.e.d.
P.: 1./T8/C, T11, T12, T14.
C.: 1./T1.
From the theorem it is clear: if YC>0 then part sums of all non-empty subclasses of cumulated yield class are also positive numbers.Q.e.d.
P.:
C.: 1./T
8.Theorem 9: In the tth time point (t=1,2,....) to the subclass of static cumulated gross result class of the net property, which named static cumulated costs class, belongs part sum, as in the tth time point existing quantity or monetary value (or its other characteristic's measure) of the cumulated costs, we may express with only negative number (T9).
Let CC denote in the tth time point (t=1,2,...) existing the size of cumulated costs, which sprang in the economist's economy in the interval (0;t]. Let us show that the part sum CC of the non-empty static cumulated costs class less then zero, that is: CC<0.
By the definition under the cumulated costs (value) in the tth time point (t=1,2,...) must be understand that decrement of the eigen capital (alias equity) which eventuated in interval (0;t] of economy, but not included the decrement of the startup and/or reserve capital. This equity's decrement, within this the increment of loss, may materialize in shape of any used of the assets, eventual payment,[49] given goods or services (to barter also) or issued liability, additionally the property's natural decrement or rather credit's release. This decrement equal with the part sum of the cumulated costs class named to final class of the equity named to static relative base class in the tth time point.
Let this equity's decrement materialize now e.g. in shape of eventual cash payment, which was executed for some received service. Let DM denote this paid cash's quantity, which decreased the supply S'M of money (by A15). S'M, as property-part, is greater then zero by T1, that is: S'M>0. Now if the cost means paying of a money-dose M, where 0<M≤S'M, then M must be detracted from S'M. Consequently the cash payment M, as negative number decreases the positive sign supply S'M of money (by A15). Thus instead of M let -M denote this cash payment, which by the definition of the cumulated costs equal to CC, that is, CC=-M (their measures are same). This time: -M<0, so CC=-M<0 and thus -M=CC<0.
We may show alike that if the cumulated costs materializes in other form of the property, in the aforesaid, then CC<0 also.
Q.e.d.
P.: 1./T8/C, T10, T11, T12.
C.: 1./A15, T1.
From the theorem 9 it is clear: if CC<0 then part sums of all non-empty subclasses of cumulated costs class are also negative numbers (T9/C).P.:
C.: T
9.Theorem 10: If in the tth time point (t=1,2,....) the cumulated or rather the current period yield less then the with it in same measure given cumulated or rather current period costs' absolute value, then the in tth time point existing cumulated or rather current period gross result's name is loss, if the yield is greater, then its name is profit, both are implicitly cumulated or rather current period (T10).
Let Y denote the measure of the cumulated or rather the current period yield just as let C denote the measure of the cumulated or rather the current period costs, item let |C| denote absolute value of C; and finally let R denote the measure of the cumulated or rather the current period result in the tth time point (t=1,2,...). (Measure of Y, C and R is same.)
Hereafter I mention briefly only yield or cost or result without notation of the season, but these belong always to same interval and may be cumulated or rather in current period.
Now let us show that
a) if Y<|C| then name of result R is loss, while
b) if Y>|C| then name of result R is profit.
By the condition of the case a): Y<|C|. But Y-|C|<|C|-|C|, however |C|-|C|=0 and thus Y-|C|<0. But C<0 by T9 and hence |C|=-C. Thus we may write that: Y-|C|=Y-(-C)=Y+C<0.
Since by the definition of the cumulated or rather the current period result: Y+C=R, just as: Y+C<0, thus R<0. But then from these and from the relating definition it directly flows that the name of the cumulated or rather the current period result R<0 is the cumulated or rather the current period loss.
By the condition of the case b): Y>|C|. But this time it suffices if in all steps of proof of the case a) we write for the relation sign '<' the relation sign '>'. From this promptly it flows: R>0. But then from R>0 and from the relating definition it directly flows that the name of the cumulated or rather the current period result R>0 is the cumulated or rather the current period profit.
Q.e.d.
P.: 1./T10/C.
C.: 1./T9.
From this theorem 10 it is already clear that in the tth time point (t=1,2,...) the cumulated or rather current period result R may be any sign number (R0) (T10/C).
P.:
C.: 1./T
10.Theorem 11: Let the gross property or its some part be classified in the tth time point (t=1,2,....) by assets- or capital-aspect. Additionally: the property changes, which ones result in this gross property or its part, let them be classified by time-aspect in the interval (0;t]. Thus in the tth time point to this static property class belonging main or rather part sum equal to the time classes of interval (0;t] belonging with sum of part sums which may be only non-negative number, except the part sum of the equity class and result class which may be any sign number, just as the part sum of the costs class which may be only non-positive number (T11).
Let CST denote an assets- or capital-aspect static property class in the time-point t=M (t,M=1,2,...). Just as let CCH denote property change class in the interval (0;M], which resulted in the interval (0;M] by the means of property change happened in the class O existing property or property hiatus in the Mth time-point.
Now let PM(CST) denote to the static property class CST in the Mth time-point belonging main or rather part sum. Additionally let P(0;M](CCH) denote the main sum of the property change class CCH which expressed by same measure as PM(CST). Finally let I(t) denote the part sum of some dynamic time class of the property changes class.
For A4 is true: P(0;M](CCH)=
I(t). Thus the formula of the theorem 11 is the following:
PM(CST)=P(0;M](CCH)=I(t)³
0 (t,M=1,2,...), except the case of the equity (aka: eigen capital), the result and the costs. Let us show this one.
Now then the followings are standing:
(1/a) PM(CST)³ 0 by T1, T2, T6, T7, T8 and A2 if CST is not the class of the equity or the result and the costs, else
(1/b) PM(CST)0 by T5 and T10/C if CST is the class of the equity or the result, just as
(1/c) PM(CST)£ 0 by T9 és A2 if CST is the class of the costs.
Since the amount PM(CST) sprang in the interval (0;M] for the algebraic sum of the property changes, hence the following is true by A5:
(2) PM(CST)=P(0;M](CCH)=I(t) (t,M=1,2,...).
But the equal value expressions may transpose, hence the (1/a), (1/b) and (1/c) inequalities and their left-side we may write thus too:
(3/a) PM(CST)=P(0;M](CCH)=I(t)³
0 (t,M=1,2,...), just as in the case of the equity or the result class:
(3/b) PM(CST)=P(0;M](CCH)=I(t)0 (t,M=1,2,...), additionally in the case of the costs class:
(3/c) PM(CST)=P(0;M](CCH)=I(t)£
0 (t,M=1,2,...), where in the case of (3/b) and (3/c) by T5, T9 and T10/C only in the relations sign is difference.
Q.e.d.
P.: 1./T12, T14, T18,
C.: 1./A
2, A4, A5, T1, T2, T5, T6, T7, T8, T9.Q.e.d.
P.: 1./T
12.C.: 1./T
11.Corollary 2: From this theorem 11 it is already clear that to the time classes
CCH(t) of the interval (0;M] (t,M=1,2,...) belonging from part sums I(t) squarely follows to the Mth time-point belonging the value PM(CST) of the static property class CST(M). However the reverse of this one is not true. But this relationship is true on the PM(CST) and its part sums of the static sub class too.Q.e.d.
P.:
C.: 1./T
11.Let CST be a static property class by the premise in the t=M time-point (M>0 and integer) which at that time is non-empty, or empty. Let Pt(CST)=PM(CST) to the CST belonging main or part sum in the Mth time-point. Now firstly (I.): let PM(CST)>0 if CST in the Mth is non-empty (by T1, T2, T6, T7, T8, T11/C1) then PM(CST)=0 if CST even then is empty (by A2). With that let us express the two cases with one relation sign. Thus PM(CST)≥0. Secondly (II.): let us analyse the theorem 12 near PM(CST)≤0 (by T9, T11/C1, A2). Only these two situations may stand by the condition.
(I.) Now let CCH denote to the interval (0;M] belonging property changes base class. These property changes in CCH were result in the CST. Let PM(CCH) denote the main sum of the CCH. Now let us divide up in the interval (0;M] existing property change class CCH by M pieces time classes. Let I(t) denote the part sum of the tth time class (t=1,2,...,K,...,M; K,M>0 and integer). Additionally let PK(CCH) be to first K time classes of the property change class CCH belonging K pieces part sums' the sum. We may write this relationship with the follow formula: PK(CCH)=
I(t), where 1≤K≤M.
I have:
(1) If PM(CST)≥0, then PK(CCH)=I(t)≥0, where 1≤K≤M.
It is clear that in case of K=M the predicate of the theorem 12 is true by the condition and the T11. That is: PM(CST)=PM(CCH)=I(t)³
0. Hence we must demonstrate the theorem 12 only on the 1≤K≤M-1 cases.
But I have:
(2) If PM(CST)≥0, then PK(CCH)=I(t)≥0 is true, where 1≤K≤M-1.
If namely (2) is not true, then
(3) near PM(CST)≥0 PK(CCH)=I(t)<0, where 1≤K≤M-1.
But by T11 PM(CCH)=I(t)³
0, if t=1,2,...,K,...,M-1,M. Thus if t=K then the follow is true:
(4) PK(CCH)=I(t)≥0, where 1≤K≤M-1.
But then thus on PK(CCH) we get two values: PK(CCH)<0 by (3) and PK(CCH)≥0 by (4). That is PK(CCH) is less then zero and is not less then zero together. But this one is not possible. One statement and its contrary together may not be true. Since statement (4) correspond the confirmed to T11 thus the statement (3), which opposed with T11, may be only false.
(II.) It is clear that the theorem 12 easy may confirm near PM(CST)≤0 too, only we must turn round the direction of the corresponding inequality signs.
Q.e.d.
P.: 1./T12/C, T14.
C.: 1./A2, T1, T2, T6, T7, T8, T9, T11, T11/C1.
Corollary: If either part sum of some cumulated part sum property classification non-negative (or non-positive) then its other part sums are also such (T12/C).
Q.e.d.
P.:
C.: 1./T12.
Theorem 13: If in tth time-point (t=1,2,..) the main or part sum of some static property class is not zero then the static property class is non-empty (T13).
CST) denote the main or part sum of some static property class in the tth time-point (t=1,2,...). P(CST)≠0 by the condition.I have: if P(
CST)≠0 then CST is non-empty in the tth time-point. If namely CST is empty then P(CST)≠0 now this one contradicts to A2, by which: "If in a given time point some static property class is empty, if and only if its main or part sum equal to zero". Thus CST may not be empty then P(CST)≠0.Q.e.d.
P.:
C.: 1./A
2.Theorem 14: Let I(t) be part sum of tth time class of to the interval (0;M] belonging (t,M=1,2,...) property changes' classification. If in the interval (0;M] occurred property changes resulted in the Mth time-point non-negative sized gross property or its in some static class extant part then, if 1≤t≤M, any part sum I(t) may be greater then zero or equal with zero. Till if 2≤t≤M, then any part sum I(t) may be less then zero, provided that absolute value of I(t) is not greater then the sum of the first t-1part sums (T14).
Let P
GR(t)=PGR(M) denote, as main or part sum, in the Mth time-point existing or then already (or yet) not existing gross property's, or in its any static class extant part's the size. PGR(M)>0 if the property or its part in the Mth time-point exists (by T1, T2, T6, T7, T8) and PGR(M)=0 if then this ones do not exists (by A2). The two expression together let PGR(M)≥0. Additionally: let PCH(M) denote this property or its part resulting in to the interval (0;M] belonging dynamic property classification's main sum (the I(t) denotes some part sum of classification).With these notations and by T1, T
2, T6, T7, T8 and T11 just as by A2 we may write thatI(t)³
0.
Let us show that if (A) is true then the part sum of any Kth time class may be the follow:
is the I(K) previous the sum of part sums of the first K-1 time class, on which one the next statement is true:
PCH
(K-1)=
I(t)³
0 by T12.L.
The (B) do not contradicts to the follow theorems T1, T
2, T6, T7, T8, T11 and to the axiom A2 that is to the formula (A), where PGR(M)=PCH(M)=I(t)³
0. Thus (B) is true.
Now yet, let us verify the part (C) of theorem.
Firstly let us postulate with the case (C) opposite that I(K)<0 if 1≤K≤M. But this time for example let even I(1)<0. But then to the first time class belonging part sum of the gross property or debt or their parts is negative, now (A) is true. What is impossible because this time I(1)³ 0 is possible only by T12.L.
I remark that this result corresponds therewith the expressive statement also that: If I(1)<0 is true then this one means that e.g. in case of property or debt we took something out of nothing, what is nonsense. Thus: any part sum I(K) may not be negative or rather the negativity may not begin with K=1 only with K=2, provided that I(1)>0 and I(1)≥|I(2)| if I(2)<0.
Secondly let us postulate that the opposite of (C) the following (C') is true so:
(C') I(K)<0 if 2≤K≤M provided that |I(K)|>P
CH(K-1) holds.But |I(K)|=-I(K) because I(K)<0.
Thus |I(K)|=-I(K)>P
CH(K-1) exists in (C'). Now let us add I(K) to the both sides of -I(K)>PCH(K-1). This time we may write:(D) I(K)-I(K)>P
CH(K-1)+I(K). Thus the left side of (D) equal with 0, the right side of (D) now equal with the sum of the first K part sums that is with PCH(K).The follow is true in point of P
CH(K):(E) PCH
(K)=I(t)³
0 (2≤K≤M) for T12.L. Now let us write for the right side of (D) the formula (E). Thus we get that:
(F) 0>PCH
(K-1)+I(K)=PCH(K)=I(t)³
0 (2≤K≤M). But (F) indicates contradiction since 0>PCH(K)=I(t)³
0 (2≤K≤M), that is
(G) 0>P
CH(K)³ 0 (2≤K≤M).In words: the (F) and (G) indicate that P
CH(K) is less then zero and that it is not less then zero together, what is clear contradiction. Since, the consequence was not incorrect and we got still on contradiction, hence undoubtedly the hypothesis (C') is false. Consequently the original statement (C) is true.[50]That is we demonstrated that the three statements (A) and (B) and (C) are correct and thus the theorem 14 is indeed true.
Q.e.d.
P.: 1./T14/C.
C.: 1./A
2, T1, T2, T6, T7, T8, T11, T12.L.Corollary: From this theorem 14 it is already clear that if in the interval (0;M] occurred property changes resulted in the Mth time-point a some static property class with non-positive sized main or part sum then, if 1≤t≤M, any part sum I(t) of this dynamic property classification may be less then zero or equal to zero. Till if 2≤t≤M, then any part sum I(t) may be greater then zero, provided that value of I(t) is not greater then the absolute value of sum of the first t-1 part sums (T14/C).
Q.e.d.
P.:
C.: 1./T14.
Let P
GR denote size of the gross property (PGR>0; T1) and let CF denote size of the foreign capital/debt (CF>0; T2). Additionally let CE denote size of the eigen capital/equity. CE=PGR-CF0 by definition and T5.
Let us show that C
E tends to minus infinite if the economist leaves to itself its the property or any part of the property.Now by A7, if the economist in some time-point tl (tl=1,2,...) leaves to itself its the property or any part of it, then its size and monetary value, but leastwise monetary value (or other positive coefficient linear transformation's value), the natural and/or the social and/or the economic environment through generated economic events to impression, with the lapse of time monotonous decreases and tends to zero.
Just as by A11, if the economist in the said time-point tl leaves to itself its the property or any part of it, then its debt' measure, the natural or the social or the economic environment through generated economic events to impression, with the lapse of time monotonous increases and tends to the plus infinite.
But from the two cross monotonity (by A
7 and A11) consequences that with the lapse of time become single such time-point tN (tl≤tN) from which or after which the CE=PGR-CF<0 and this negativity, with the lapse of time, monotonous increases, that is CE tends to the minus infinite (CE ® -¥ ).(Remark: the existence of the more time-point is excluded for the two cross monotonity, by A
7 and A11.)Q.e.d.
P.: 1./T15/C.
C.: 1./A7, A11, A12, A13, T1, T2, T5.
Q.e.d.
P.: 1./T29.
C.: 1./T
The structural laws of the property and the property classification systems
Theorem 16: ==...=≥0, that is: if we classify to the interval (0,t] belonging property changes base class and/or to the tth time-point (t=1,2,...) belonging balance class of it, n ways (n³
2), namely by arbitrary but differing property aspect A1,A2,..,An, or if we complete the property classification system with a classification by aspect An+1, then the structures of classifications of this property classification system are differing, while the main sums of it, which ones expressed by same measure, are all equal (T16).
Let A
i be (i=1,2,...,n,n+1) an arbitrary property classification aspect and let SAi denote a part sum by some aspect Ai. Additionally let us denote the structures by Ai of classification of in the interval (0;t] happened property changes and/or their to tth time-point belonging balances the sum of the part sums expressing with follow formulas:(1)
,
,...,
,...,
, ill. 
where x,y,u,w ,x >0 and are integers. With under the (1) listed formulas symbolized structure of property classifications is all differing because by the axiom A
6: totality of given property changes or rather in a given time existing property has not two same property classifications.I have that:
(2)
=
=...=
=...=
≥0 is true.
Let SAi
denote the main sum
, that is: SAi= by A4. Thus on all i is true:
(3) SA1
=
, SA2=
,...,SAi=
,......,SAn
, illetve SAn+1=
.
Now we can write the (2) formulas easier too:
(4) S
A1=SA2=...=SAi=...=SAn≥0. Let us show that this statement is true.Firstly:
(I.) If n=2 then (5) S
A1=SA2≥0. Let us verify this statement.Now let P
GR denote the main sum of dynamic classification (which is by any aspect) of in the interval (0;t] happened gross property changes. Just as let P'GR denote the main sum of to the tth time-point belonging some static classification (which is by any aspect) of balances of in the interval (0;t] happened gross property changes.P
GR=P'GR by axiom A5, independently from it that the main sum PGR is main sum of dynamic or static classification, and from it also that what property aspect of the classification.In the tth time-point with main sum P
GR given size of the gross property, by T1 and A2, is not less then zero (PGR≥0). However by A4 the following statements hold: PGR=SA1 and PGR=SA2, hence SA1=SA2≥0 is true. Thus if n=2 then (5) and so (4) and (2) is true also.(II.) Let us postulate that is true the formula on n term, see in statements (2) and (4), and let us demonstrate that with the (n+1)th term completed formula holds too.
So, let us complete the equality-chain in (4) from (3) taken with left side of formula SAn+1
= (A4), which equal to PGR and its aspect An+1 and its class structure is new (by A6).
Let us prove that the following inequality holds:
(6) S
A1=SA2=...=SAi=...=SAn=SAn+1≥0.Now, by the premise just as T
1 and A4, the next inequality holds: VBR=SAn+1≥0.But V
BR=SAn≥0 is true also by the premise just as T1 and A4, A5.However so S
An=SAn+1. But by the premise SAi=SAn≥0 also is true (i=1,2,...). Additionally since SAi=SAn and SAn=SAn+1 hence on all i is true: SAl=SAn+1≥0 (i=1,2,...,n).Consequently the formula (6) and thus the (2) and (4) any n and n+1 term in case is true.
Q.e.d.
I name this theorem the n-aspect (n≥2) structural law of the property.
[51]P.: 1./T
16/C1, C2, C3, T17, T17.C.: 1./A
4, A5, A6, T1.Q.e.d.
P.: 1./T
C.: 1./T
16. i=CE+CF≥0, that is: if set of to the interval (0,t] belonging gross property changes by time, when the set of to the tth time-point belonging balances (that is objects of the property) by assets and capital, that is together: it is classified by three differing aspect, then the structures of classifications of this dynamic and static property classification system are different, but the in same measure expressed three main sums are equal. (T16/C2).Q.e.d.
P.: 1./T
C.: 1./T
16.Corollary 3: TM=AM=CM=...=XM≥0, that is: if the set of to the interval (0,M] belonging gross property changes by time, when the set of to the Mth time-point belonging balances (that is objects of the property) by assets and capital plus other aspect, that is together: it is classified by N differing (N³ 3 and integer) aspect, then the structures of classifications of this dynamic and static property classification system are different, but the in same measure expressed main sums all are equal. (T16/C3).
Q.e.d.
P.: 1./T
C.: 1./T
16.Remark:
We may write this corollary T16/C3 with mathematical notation thus too:
T
M=AM=CM=...=XM≥0, whereÈ È È ... È
the different aspects: 1, 2, 3, ...,N (N and integer)
and let them symbolize the classifications with the follow formulas:
I(t)≥0 and (T is denotes 'main sum of the classification by Time')
Ai≥0 and (A is denotes 'main sum of the classification by Assets')
≥0.
I name by the formula TM
=AM=CM=...=XM≥0 expressed corollary the N-aspect (N≥3 and integer) dynamic and static structural law of the property. [52]Ai(t)=CE(t)+CF(t)≥0, that is, to the tth time-points (t=1,2,...,M) belonging the same unit and A-C-aspect[53] main sums of TA-TC-aspect dynamic property classification system of the gross property and their to Mth time-point summarized sums are equal (T17).
The following formula (1) holds on the main and part sums of the A-C-aspect static classification of the gross property in some tth
time-point by T16/C1:(1) Ai
Therefore, we may sum to the all intervals (t-1;t] belonging A-C-aspect changes of the gross property.
I have that the following formula (2) holds in view of the TA-TC-aspect dynamic classification system of changes of the gross property according to this theorem,
(2)
Ai(t)=CE(t)+CF(t)≥0, just as on any tth term of two sums of the equality-part of (2) the following formula
(3)
Ai(t)=CE(t)+CF(t) (t=1,2,...,K,...,M) also holds (see the table T18.)
|
4 aspect complex dynamic and static balance sheet | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
|
Time-assets dynamic classification |
Static classification | ||||||||||
|
1 |
2 |
3 |
|
|
|
4 |
|
|
|
|
|
5 |
|
Assets-types |
|
Times (days) |
a(i)s |
|||||||||
|
|
a |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
|
Sheep |
1 |
1 000 |
10 |
-10 |
|
|
|
|
|
|
|
1 000 |
|
Cow |
2 |
2 000 |
|
|
|
|
|
|
|
|
-1 000 |
1 000 |
|
Feed |
3 |
|
|
|
|
|
|
|
|
600 |
|
600 |
|
Claims from buyers |
4 |
300 |
|
|
-200 |
|
|
|
|
|
700 |
800 |
|
Gold money |
5 |
|
|
|
200 |
|
50 |
150 |
1 100 |
-600 |
700 |
1 600 |
|
p(t) |
|
3 300 |
10 |
-10 |
0 |
0 |
50 |
150 |
1 100 |
0 |
400 |
5 000 |
|
Cumulated p(t) |
|
3 300 |
3 310 |
3 300 |
3 300 |
3 300 |
3 350 |
3 500 |
4 600 |
4 600 |
5 000 |
|
|
|
||||||||||||
|
|
Time-capitals dynamic classification |
Static classification | ||||||||||
|
1 |
2 |
3 |
|
|
|
4 |
|
|
|
|
|
5 |
|
Capitals-types |
|
Times (days) |
c(j)s |
|||||||||
|
|
c |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
|
Cumulative result |
1 |
2 700 |
|
|
|
|
|
|
|
|
|
2 700 |
|
Services income |
2 |
|
|
|
|
|
50 |
150 |
|
|
|
200 |
|
Revenue |
3 |
|
|
|
|
|
|
|
|
|
1 400 |
1 400 |
|
Wagework revenue |
4 |
|
|
|
|
|
|
|
1 100 |
|
|
1 100 |
|
Accretion |
5 |
|
10 |
|
|
|
|
|
|
|
|
10 |
|
Labour cost |
6 |
|
|
|
|
|
|
|
|
|
-100 |
-100 |
|
Goods cost |
7 |
|
|
|
|
|
|
|
|
|
-1 000 |
-1 000 |
|
Long-term liabilities |
8 |
500 |
|
|
|
50 |
|
|
|
|
|
550 |
|
Current liabilities |
9 |
100 |
|
-10 |
|
-50 |
|
|
|
|
100 |
140 |
|
Opener balance |
10 |
|
|
|
|
|
|
|
|
|
|
0 |
|
p(t) |
|
3 300 |
10 |
-10 |
0 |
0 |
50 |
150 |
1 100 |
0 |
400 |
5 000 |
|
table 18 |
||||||||||||
Let us verify these two statements (2) and (3).
Firstly:
The two main sums
Ai(t) and CE(t)+CF(t) in (2) represent the two different classification of the gross property by the premise and the concerning definitions just as A6. Additionally, these main sums, one by one, equal to the main sum of the gross property by mentioned definitions. For these two reasons the formula (2) evidently is same with the general formula of T16 in case n=2.
Secondly:
Let us verify yet the statement (3).
If t=1 then (3) is true because the (2) near t=1 also is true, and this time (2) and (3) are identical.
But the formula (2) is true near t=K and t=K+1 too (K=1,2,...,M). However, if (2) is true near t=K+1 also then the following formula (4) is true also:
(4) Ai
(K+1)+
Ai(t)=CE(K+1)+CF(K+1)+CE(t)+CF(t)
because the formula (4) is identical with the next equality: 
Ai(t)=CE(t)+CF(t) which is true by T16 if n=2.
But if (4) is true then the following formula (5) is also true:
(5) Ai
So (5) in case t=K+1 (t=K+1=2,...,M) and (3) in case t=K (t=K=1,2,...,M) are true and thus the theorem is true too.
Q.e.d.
P.: 1./T
17/C.C.: A
6, A7, A11, A15, T16, T16/C1.I name this theorem 17 to time-assets and time-capitals-aspect dynamic structural law of the gross property.[54]
Corollary: The to the tth time-point (t=1,2,...M) same unit main sums of two arbitrary two different aspect dynamic property classifications of the gross property and to Mth time-point summarized their sums are equal (T17
/C).Q.e.d.
P.:
C.: T
17.I name with this corollary expressed statement the arbitrary two-aspect dynamic structural law of the gross property.
[55]I(t)=Ji(t)=VS(t)+VI(t)≥0, that is, to the tth time-points (t=1,2,...,M) belonging the same unit and A-C-aspect[56] main sums of T-TA-TC-aspect dynamic property classification system of the gross property and their to Mth time-point summarized sums are equal (T18).
Let us see the following formula:
(1) V(M)BR
=VBR(t)=I(t)≥0 (T1, A2),
where V
BR(t)=I(t) (t=1,2,...,M) just as VBR(t) and I(t) denote the same part sums of the tth time classes. The formula (1) in turn, by definition, is a with main sum and part sums expressing mathematical formula, which represents a clear by time-aspect classification of to the interval (0;M] belonging changes of the gross property.Let T-TA-TC-aspect denote hereafter briefly the following expression: 'time-time and assets-time and capitals-aspect'.
I have by these that the following T-TA-TC-aspect formula, which corresponds to statement of the theorem, is true:
(2) I(t)=
Ai(t)=CE(t)+CF(t)≥0 and the any tth terms of the sums of equalities (2) are true also (see below):
(3) I(t)=Ai
Let us verify these formulas (2) and (3).
Firstly: The I(t) and
Ai(t) just as CE(t)+CF(t) all represent, one by one, different classification in this property classification system by the condition and the A6. Additionally, these sums by the condition, one by one, are equal with the main sum of the gross property. Hence the formula (2) clearly same with the general formula of the T16 if n=3. Therefore this part of the theorem is true.
Secondly:
Now yet let us verify the formula (3) validity [(3) I(t)=Ji
In the formula (3) in equality Ji
Ji(t) and the sum VS(t)+VI(t) both algebraic sum of in the tth time class belonging property changes, or other: algebraic sum of in the interval (t-1;t] belonging property changes. Hence these are equal with the part sum I(t) too.
So (2) and (3) are true, thus the theorem is true too.
Q.e.d.
P.: 1./T
18/C1, C2, T20.C.: 1./A
6, T16/C2.Corollary 1: The to some tth time-point (t=1,2,...M) belonging part sum of time-aspect property classification of the gross property equal with in the same unit expressed main sum of by any other but time and another aspect property classification which main sum is also to this tth time-point belongs (T18/C
1).Q.e.d.
P.:
C.: T18.
1).Q.e.d.
P.:
C.: T18.
The relations of the economic events and the property classification systems
Theorem 19: Any and however many economist-specific economic event also occurs this fact does not affect the validity of T-A-C-aspect[57] dynamic and static structural law of the gross property while this time to the economic event-coordinates corresponding to the final property class belonging part sums change to the character of the economic event(s) accordingly.
Let PGR(t) denote the main sum of in the time changing gross property of the economist (t=1,2,...,M). Let the next formula show in the t=0 time-point inventoried gross property measure and structure: P
GR(0)=I(0)=
Ai=CE+CF≥0 where the terms of the inequality are expressed by quantity or monetary value or their other positive coefficient linear transformed.
Now let us investigate the conformation of validity of T-A-C-aspect dynamic and static property structural law in the interval [1,M] occurred apropos of economic events (by A
12, A13) the by T16/C2 the following formula with use:(1) I(t)=Ai
In this the I(t) denotes the part sum of tth time class of gross property changes.
Thus we must show that the validity of the inequality in (1) any and however many economist-specific economic event (by A
12, A13) occurs it remains true, while this time to the economic event-coordinates corresponding to the final property class belonging part sums change to the character of the economic event(s) properly.By the axiom A
15: In some tth time point (t=1,2,...) occurred apropos of economic event into the touched property classification (a) only final class' part sum increase with an amount D X (D X>0), or (b) decrease with D X (to the decrease, let c denote, holds: c=-D X<0), or (c) either final class' part sum decrease with D X, while either other final class' part sum increase with the selfsame D X to the (t-1)th time point compared. Other character elemental property change apropos of economic event or from other cause (by T12) is not possible.So we must investigate only the effect of three kinds changes the by A
15.We must show that any possible property change bearer economist-specific economic event does not make invalid the formula in (1). As in (a) and (b) the sign of change is only different therefore their impact can be tested altogether too. Let this one be the case (A) while let case (B) be the type (c) of the changes.
Let it denote by the (A) apropos of some economic event occurring change of gross property:
D PGR=(PGR± x)-PGR=± x, where clear x>0 for T1.
Additionally, we bear on simpler format the equity and foreign property classes with additional classification. Let C
E,w and CF,p be to the final capital classes belonging two part sums, where the two index w and p are both positive integer. This time:(A1) C
E=CE,1+...+CE,w+...+CE,k (CE,w is part sum of an equity class),(A2) C
F=CF,1+...+CF,p+...+CF,r (CF,p is part sum of a debt class).End if, for reasons of convenience, we sum the maximal values of the two indexes w and p and then for C
E,w and CF,p we may introduce the following general capital variables: Cj where j=1,2,...,z=k+r.This we write:
(A3) PGR(t)=C
E+CF=C1+...+Cj+...+CZ, or briefly:(A4) PGR(t)=C1+...+Cj+...+CZ=
Cj.
The formula (1) now possible transformed with the new notations so:
(2) I(t)=Ai=Cj³
0 which in following may detail is such:
(A5) I(1)+...+I(t)+...+I(M)=A1+...+Ai
+...+An=C1+...+Cj+...+CZ³ 0.Now: by the premise, let an economic event occur in some tth time-point of some interval [1;M] and the size D PGR of the thus resulted property change let D PGR=± x (x>0; T1). This economic event may occur only the economist's economy activity or the natural or the social or rather the economic environment to impression (by A
12). And let this event touch for example the assets ith class, that is its the part sum Ai (Ai± x). This time the T-A-C-aspect main sums of this gross property must agree with each other by A4 and T16/C2. But these main sums only then may be equal if the sides of equality-part in (3) increase with same sum or rather if those decrease with same sum. Or other: if the part sum I(t) of a time-type and the part sum Ai of a asset-type just as the part sum Cj of a capital-type change all at once by same measure and sign. (Or else A4 and T16/C2 would be false, in turn both are true.)Now let C
j be so the changing part sum of the capital-type and let I(t) be the changing part sum of the time-type. Thus really the equality-inequality will not invalid only the corresponding three part sums change. So the formula in (A5) thus forms:(A6)
I(1)+...+[I(t)± x]+...+I(M)=A1+...+(Ai± x)+...+An=C1+...+(Cj± x)+...+CZ³ 0.[58]Thus the validity of the equality-inequality spite of the property change D V
BR=± x (x>0) remained because all three aspect main sums changed identically with value ± x. We may show this fact underlined the ± x below:(A7) ±
x+I(t)=±
x+Ai=±
x+Cj³
0.
Naturally, the property change just therefore happened because by premise the economic event may have occurred, that is, that was not an impossible economic event by definition.
Namely, by the event coordinate triple <t,i,j>,[59] the part sum of the touched property classes should not change sign if it was not possible about character of the property class or the economic event. The asset-aspect part sum A
i may be zero but may not be negative because we may not took something out of nothing; namely the opposite would be nonsense. (For example: if there is not a penny nor in the checkout then the poor cashier has not to give thereof nothing; the opposite would be nonsense.)While if C
j is the part sum of the costs class then it may be zero but may not be positive because then the property such decrease for cost that increase; or other: then the property such decrease for cost that the cost is yield by definitions. But all these are nonsense. (For example: if the part sum Cj of the costs class greater then zero then this means that the elements of the costs class that is the missing (the lost) assets become existing assets, or other: the word 'none' means that 'there is a'. But clear this is nonsense.)We exploit that D PGR=± x (x>0 for T1) can be any frequent and any large increase or decrease, near the foregoing non-negativity or rather non-positivity limit, because we do not define the size and the frequency of change D PGR=± x. We exploit also that t may be any time-point in the interval [1;M] and M may be any large integer. Additionally, we exploit also that D PGR=± x may be added to any I(t) and to any A
i (i=1,2,...,n) or rather to any Cj (j=1,2,...,z), provided that the economic event coordinate triple <t,i,j> is realistic by definition.[60]On the basis of these we can see that any number of such type and arbitrary economic event may occur and cause arbitrary size property change D PGR=± x (x>0) this one does not void the formula (2). With this we verify the theorem on the case (A).
Now we verify the theorem on the case (B) too. Let us go from the formula (A5) again. But near the general element A
i among those let us do the other general element Ak too (where 1£ i,k£ n and i¹ k, provided that <i,k>[61] economic event coordinates are realistic):(B1) PGR(M)=I(1)+...+I(t)+...+I(M)=
A1+...+Ai
Firstly, let them change those part sums which ones belong to the apropos of in tth time-point occurred economic events touched A-aspect final classes. Let them change with a value x>0 by part (c) of the axiom A
15.If this resulting in change economic event occurred and let us suppose that in the tth time-point the part sum A
k of an asset-types decrease with x, that is: A'k=Ak-x≥0, while an part sum Ai of an other asset-types increase with x, that is: A'i=Ai+x≥0 (provided that economic event coordinates <i,k> are realistic). This time we write:A'i+A'k=(Ai+x)+(Ak-x)=(x-x)+Ai+Ak=0+Ai+Ak=Ai+Ak.
So the sum of part sums of the ith and kth assets classes did not change (A'i+A'k=Ai+Ak) because the opposed changes compensated each other, namely x-x=0, that is in this case, in total, change is not. Consequently this time the main sum of the assets classification also can not change for A
4.But this time the part sum I(t) of to the tth time-point belonging interval (t-1;t] that is time class such change that at once increase and decrease also with x, so in truth it is not change.
That is the formula (B1) will be thus:
(B2) PGR(M)=I(1)+...+[I(t)+x-x]+...+I(M)=
=A1+...+(Ai+x)
+...+(Ak-x)+...+An=C1+...+Cj+...+CZ³ 0,or other: since x-x=0, hence
I(t)+x-x=I(t)+0=I(t), that is
(B3) PGR(M)=I(1)+...+I(t)+...+I(M)=
=A1+...+A'i
+...+A'k+...+An=C1+...+Cj+...+CZ³ 0,where A'i=Ai+x and A'k=Ak-x are the new values of the part sums Ai and Ak (1£ i,k£ n; i¹ k; provided that the economic event coordinates <i,k> are realistic). If we regroup the terms +x and -x on one place then we can exemplify the cause of unchanged of the main sum:
(B4) PGR(M)=I(t)
Ai)+x-x=Ai=
Cj³
0,
since x-x=0, and (Ai)+x-x=(Ai)+0=Ai.
The T and A and C-aspect main sums, in the equality-inequality system (2) and (B1), did not change despite the size x>0 and assets-aspect structural-change.
To verify this theorem we exploit that this structural change can be any frequent and that the value x (x>0) of the structural change can be any large, near the foregoing non-negativity limit, because we do not define the size and the frequency of change x. We exploit also that t may be any time-point in the interval [1;M] and M may be any large integer. Additionally, we exploit that x may be added to any Ai and may be deductible from any Ak (i,k=1,2,...,n; but i≠k), provided that the economic event coordinate pair <i,k> is realistic by definition.[62]
We may show on analogous method that the formula (2) holds then also if the structural property change x>0 occurred on the capital classes, thus:
(B5) PGR(M)=I(t)=Ai=C1+...+(Cj+x)+...+(Ch-x)+...+Cz=(Cj)+x-x³
0,
[where (1£ j,h£ z; j¹ h), provided that the pair < j,h> is realistic) or rather then also if the structural property change x>0 occurred only on the time classes, although this latter may not be valid economic event. Here it also holds:
(B6) VBR=I(1)+...+[I(t)+x]+...+[I(u)-x]+...+I(M)=[I(t)]+x-x=
=[I(t)]+0=I(t)=Ai=Cj³
0
where (1£ t,u£ M; t¹ u), provided that the coordinate pair < t,u> is realistic.
With this we verify the theorem on the case (B) too.
Since the (A) and (B) are true hence the theorem 19 is also true.
Q.e.d.
P.: 1./T19/C1, C2; T21.
C.: 1./A2, A4, A12, A13, A14, A15, T1, T16/C2.
Corollary 1: Absolute or relative main sum of any property classification is covariant (it changes same way) apropos of economic event occurring in point of increase or decrease of its part sum, while invariant in point of compensatory (an opposite sign but equal in size) changes of its two part sums.
P.: 1./T
28, 2./T8.C.: 1./T
19.P.:
C.: T
19.Theorem 20: Any and however many economist-specific economic event also occurs this fact does not affect the validity of T-TA-TC-aspect[63] dynamic structural law of the gross property while this time to the economic event-coordinates corresponding to the final property class belonging part sums change to the character of the economic event(s) accordingly.
If we make the variables A
i,CE,CF of the formulaI(t)=
Ai=CE+CF³
0 of T16/C2 into dependent from t, when to all t the interval (t-1;t] and, by definition, balance I(t) of in the interval (t-1;t] occurred property changes belongs, then we get the following formula-line:
t) tth formula
1) I(1)=Ai
2) I(2)=Ai
.
t) I(t)=Ai
.
M) I(M)=Ai
If we summarize the corresponding terms of formulas then we get the formula (2) of theorem T
18 which is verified [let us denote now with (A)]:(A) I(t)=A
CE(t)+CF(t)³
0.
Now we must show that any and however many (but it is not impossible) economist-specific economic event also occurs, this fact does not affect the validity of T-TA-TC-aspect dynamic structural law of with the formula (A) represented gross property while this time to the (realistic) economic event-coordinates corresponding to the final property class belonging part sums change to the character of the economic event(s) accordingly.
But A
Ai and CE(t)+CF(t)=CE+CF=Cj if we summarize firstly only by time (by t) the changes of the asset-types and capital-types while by i and j we don't. But this time formula (A) is equivalent with formula (B):
(B) I(t)=A
Cj³
0. However (B) is equivalent with the formula I(t)=Ai=CE(t)+CF(t)³
0 of theorem T19. But the statement of T19 is same with the statement of this theorem and T19 is already proofed. Consequently this theorem is true too.
Q.e.d.
P.: 1./T
20/C1, C2, C3, C4, C5, C6, C7.C.: 1./T
16/C2, T18, T19.Corollary 1: The material position of the economist and its all factors apropos of economic-specific economic events in the time change.
P.: 1./T
29.C.: 1./T
20.Corollary 2: The classifications of with the formula T=A=C≥0 represented property classification system independent each from the other regarding on the only structural property changes.
P.:
C.: 1./T
20.Corollary 3: In the classifications of with the formula T=A=C≥0 represented property classification system, by the characteristic of the system, if the property increase or decrease then always 3 part sums change, one in the T and one in the A and one in the C-aspect classification, while if only the structure of some classification change then always 2 part sums change but only in T or only in A or only in F-aspect classification.
P.:
C.: T
20. In the with formula T=A=C=...=X³ 0 represented N aspect (N³ 3 and integer) property classification system, by the characteristic of the system, if the property increase or decrease then always N part sums change, but in a classification only one, while if only some classification structure changes when any classification is independent from the others, then always 2 part sums change in the either classification. If in the system there are yet additional non-independent K (1≤K≤N-3 and integer) property classification then at the most 2K+2 part sums changes in all.P.:
C.: T
20.Corollary 5: Aside from the time-aspect, in with the formula A=C³ 0 represented property classification system, by the characteristic of the system, apropos of any economic event always only 2, to the A and/or the C property classification belonging, part sums change, if the property soever changes too.
P.:
C.: T
20.Corollary 6: The structural law of with the formula T=A=C=...=X=0 represented explicit N-fold (N³ 3 and integer) dynamic and static or with the formula TA=TC=...=0 represented implicit N-fold (N³ 2) dynamic property classification system will hold without the property and debt beginner economist in case (P
GR=0 and CF=0), just as if the economist has only debt (PGR=0 and CF=D>0 és CE=-D<0 and C=CE+CF=0), any and however many (but it is not impossible) economist-specific economic event occurs.P.: 2./T
2.C.: 1./T
20.The law of the natural property classification and the natural property classes
Corollárium 7: In the tth time-points (t=1,2,...,M) occurring economist-specific economic events e
i(t) [i=1,2,...,n] gradually and by natural chronology build up and in all tth time-points clearly define the property classification system of the economist. In all tth time-point of this natural process those part sums which ones correspond the character and coordinates of the events ei(t) change, that is, increase and/or decrease. This one happens then also if these changes are recorded and then also if they are not; and then also if coordinates of these events yet only can be inferable from the dates of time-point and name (description) of the economic events.P.:
C.: T
20.I name this theorem T
20/C7 to the law of the natural property classification, while the sprung classes to the natural property classes.
Complete and incomplete property classification systems
Theorem 21: With the formula T=A=C≥0 represented explicit N-fold (N=3) property classification system of in the interval (0,M] changing gross property is complete system (T
21).By the respecting definition: I name to complete a property classification system if it is satisfactorily informative and closed in point of the economist-specific economic events.
We must show that with the formula
(1) I(t)=Ai=CE+CF³
0 (by T16
represented property classification system satisfactorily informative and closed in point of the economist-specific economic events, that is, complete system.
(A) Firstly let us show that with the formula (1) symbolized property classification system is satisfactorily informative. This one holds if the formula (1) which represents the property classification system shows the economist's material position in a given time and at least the changes of its gross property to this time by the time-aspect property classification, per definiendem.
But under the economist's material position I understand the size of the gross and net property just as liability (aka: foreign property or all sorts of debt), at a given time; additionally structures of its property's classes and part sums.
In the formula (1) at the Mth time-point summed gross property P
GR≥0 (T1, A2) performs with its assets (Ai) and capitals (CE+CF) aspect main sums [see the following formula (2)]:
(2) PGR
=Ai=CE+CF³
0.
Additionally, in (2) the equity (3) C
E0 (by T5) also performs, just as the debt (foreign capital) (4) CE≥0 (by T2, A2) too; and we can see the structures of these also, as:
(5) PGR=Ai=CE+CF=A1+..+Ai+..+An= CE+CF³
0.
Thus the formula (1) shows the economist's material position in the Mth time-point, and its first term shows the changes of the gross property to the Mth time-point in the interval (0;M]; see below the formula (6):
(6) PGR=I(t)³
0.
So, with the formula (1) represented property classification system is really satisfactorily informative by the formulas (2), (3), (4), (5) and (6).
(B) Now yet we must show that with the formula (1) symbolized property classification system is closed in point of the economist-specific economic events.
I name to closed the economist's property classification system in point of the economist-specific economic events, if and only if any so economic event results there are such part sums in the property classification system which correspond to character of event and change by the content of event, per definiendem.
By the proved theorem T19: Any and however many economist-specific economic event also occurs this fact does not affect the validity of T-A-C-aspect[64] dynamic and static structural law of the gross property while this time to the economic event-coordinates corresponding to the final property class belonging part sums change to the character of the economic event(s) accordingly. This law is represented with the following (T) formula:
(T) PGR=I(t)=Ai=CE+CF³
0 (by T19).
The formula (T) is same with the formula (1), consequently with the formula (1) represented property classification system is closed in point of the economist-specific economic events.
So, we showed that with the formula (1) represented property classification system is satisfactorily informative (by A) and closed in point of the economist-specific economic events (by B), that is, complete system.
Q.e.d.
P.: 1./T21/C1, C2, C3, C4, C5; T23, T24, T28.
C.: 1./A
2, T1, T2, T16/C2, T19.Corollary 1: With the formula T=A=C=...=X³ 0 represented explicit N-fold (N³ 3) property classification system of the gross property is complete.
Q.e.d.
P.: 1./
T23, T24, T25.C.: 1./T
21.Corollary 2: With the formula TA=TC=...=TX³ 0 represented implicit N-fold (N³ 2) property classification system of the gross property is complete.
Q.e.d.
P.: 1./
T23, T24, T25, T26, T28.C.: 1./T
21.Corollary 3: If the classification system of the gross property consists (possibly near other static property classifications) only from T, or A, or C, or A and C, or T and A, or T and C-aspect property classification, or it does not contain one of these nor, then such property classification system incomplete, although with the formula A=C³ 0 represented property classification system is closed in point of the economist-specific economic events.
Q.e.d.
P.: 1./
T26, T28.C.: 1./T
21.Corollary 4: The time-, asset- and capital-aspect and by T-A-C-aspect property classification is immanent feature other attribute of the property classification.
Q.e.d.
P.:
C.: T
21.Corollary 5: The maximal number of the authoritative property-aspects is n, and 3<n<X(t,A), where X is natural number; its size is unknown and the value of the upper limit depend from the t time-point (t now is expressed in calendar year) and from the economic profile of the economist, just as from size and complexity of its economy, to which we may characterize the structure and main sum (A=å ai) of assets.
Q.e.d.
P.:
C.: T
21.Theorem 22: With the formula T
M=AM=CM≥0 or the AM=CM≥0 represented[65] property classification system, which shows in cash-flow aspect from the gross property only the money property, is incomplete.Let it represent the explicit N-fold (N=3) property classification system of the gross property [PGR≥0 (T1, A2)] the formula PGR=I(t)=Ai=CE+CF≥0. This system is complete (by T21
Let P
MGR denote from the size of the gross property the size of the money property PMGR≥0 (T1, A2), while the size of the non-money property PNGR≥0 (T1, A2)[66].This time clearly PGR
=PMGR+PNGR=I(t)=Ai=CE+CF≥0, where(M) PMGR
IM(t)=AMi=CME+CMF≥0 and
(N) PNGR
=IN(t)=ANi=CNE+CNF≥0.
Let us investigate with the formula TM
=AM=CM≥0 represented property classification system of the money property.Let us suppose that with the formula
(M) PMGR
=IM(t)=AMi=CME+CMF≥0 represented property classification system is complete.
Let us attempt to proof this statement.
(A) The complete property classification system is satisfactorily informative and closed in point of the economist-specific economic events, by definition. But P
MGR is not same the size of the gross property, because such economic event is always in line of the economist-specific economic events which does not touch the economist's monetary instruments or its other money property (by A16) only other part of the property. Thus the effect of such economic event can not show with the formula TM=AM=CM≥0 represented in property classification system, in which there are not to this event corresponding final classes. For this it is true that: PGR>PMGR, CE>CME, CF>CMF; that is: PMGR, CME, CMF do not contain the total property, the total equity or rather the total debt/foreign capital. Namely the sums of the non-money property and its sources miss from these. Consequently, PMGR, CME, CMF can not inform us on the non-money property (PNGR), on its equity (CNE) and its foreign capitals (CNF). Hence the property classification system by (M) can not be complete.(B) Additionally, PGR
>IM(t) also clearly holds, because IM(t) shows only the conformation of the gross money property, but the conformation of the non-money property does not (A16). But we had that with the formula TM=AM=CM≥0 represented property classification system is complete, that is: it inform on the gross property, however all this is not true. So we received a contradiction. Consequently, to the (M) belonging statement is false but the opposite of it is true.
Thus we proved in the parts of (A) and (B) that with the formula (M) represented property classification system is incomplete.
(C) Now we can show also that with the formula (M) represented property classification system is not closed in point of the economist-specific economic events. Namely it is not able to show the effect of economic events that change the non-money property's main sum and/or its part sums, since there is not single final class nor in the TM
=AM=CM≥0 represented property classification system with the TN=AN=CN≥0 represented system's from part sums. However, such economic event is always in line of the economist-specific economic events; this does not touch the economist's monetary instruments or other: its other money property (by A16) only other part of the gross property. But effect of this clearly does not appear in by (M) property classification system. Consequently, such system does not satisfy the other condition of the completeness nor, that is, it is not closed in point of the economist-specific economic events. This system therefore is not complete this time nor.(D) Finally: with (M2) PMGR
=AMi=CME+CMF≥0 formula represented property classification system incomplete for (A) and (C) too, but it is incompletion of (M2) yet that by time change of the money property it does not show.
Thus we verified by (A), (B), (C) and (D) that: With the formula TM=AM=CM≥0 or the AM=CM≥0 represented property classification system, which shows by cash-flow aspect from the gross property only the money property, is incomplete.
Q.e.d.
P.:
C.: 1./A
2, A16, T1, T21.Theorem 23: If the property classification system of the gross property is complete, is therein time, assets and capitals classification.
The property classification system of the gross property denoted PGR [PGR³ 0 (T
1,A2)] is complete by the condition. Now let us suppose that (H) is not therein time, assets and capitals classification.But this hypothesis (H) contradict the already proved to theorems T
21, T21/C1 and T21/C2 that after in a complete system is time, assets and capitals classification. Thus the statement (H) is false; consequently the theorem 23 is true.Q.e.d.
P.:
C.: A
2, T1, T21, T21/C1, T21/C2.
The property classification system of the gross property denoted PGR [PGR³ 0 (T1,A2)] is complete by the condition. Now let us suppose that (H) it is not closed in point of the economist-specific economic events.
But this hypothesis (H) contradict the already proved to theorems T
21, T21/C1 and T21/C2 that after a complete system is closed in point of the economist-specific economic events.Thus the statement (H) is false; consequently the theorem 24 is true.
Q.e.d.
P.:
C.: A
2, T1, T21, T21/C1, T21/C2.Theorem 25: The N-pan (N≥2) balance sheet of the gross property is complete system.
With the formula T(M)=A(M)=C(M)=...=X(M)³ 0 represented explicit N-fold (N≥3) or rather with the formula TA(M)
=TC(M)=...=TX(M)³ 0 represented implicit N-fold (N≥2) property classification system of the gross property is complete [by T21/C1, T21/C2].Now by the relating definition: we name to N-pan or other to N-fold balance sheet (N³ 2) the gross property's implicit N-fold/N-entry (N³ 2) or explicit N-fold/N-entry (N³ 3) property classification system (in some Mth time-point). Thus by this definition the N-pan (N≥2) balance sheet of the gross property is explicit or implicit N-fold/N-entry (N³ 2) property classification system, and as such is complete system.
Q.e.d.
P.:
C.: T
21/C1, T21/C2.Theorem 26: If a property classification system is complete, then it is an explicit N-fold (N³ 3) system, and at least a dynamic T, a static A and a static F property classification perform between its classifications, or it is the implicit N-fold (N³ 2) system, and at least a dynamic TA and a dynamic TC complex property classification perform between its classifications.
Let S denote a property classification system, and N the number of its aspects; additionally let them denote the dynamic time, the static assets and capitals, just as the dynamic time-assets and time-capitals aspect classifications one by one: T, A, C, or rather T-A and T-C. And if the property classification system is explicit N-fold (N≥3) then let us denote this with S
expl és Nexpl, while if it is implicit N-fold (N≥2) then let us denote this with Simpl és Nimpl.Let us verify the theorem the explicit N-fold (I.) and the implicit N-fold (II.) onto cases separately.
(I.) This time, if we slightly reword this part of the theorem and use the introduced notations, we may write:
(1) For all complete S
expl is true that: Nexpl≥3 and is therein T and A and C property classification.Let us suppose that the opposite of (1) is true, that is:
(2) For all complete S
expl is not true that: Nexpl≥3 and is therein T and A and C property classification. Or other: There is such complete Sexpl on which is true that: or Nexpl<3 and/or is not therein T and/or not A and/or not C property classification. (Here e.g. 'T and/or A' means that 'T or A, or T and A'.)This time in the complete S
expl let Nexpl<3 and let its property classification system consist (possibly near other static property classifications) only from T, or A, or C, or A and C, or T and A, or T and C-aspect property classification, or it does not contain one of these nor. Or: let Nexpl≥3, but let the complete Sexpl consist (possibly near other static property classifications) only from T, or A, or C, or A and C, or T and A, or T and C-aspect property classification, or it does not contain one of these nor.And by T
21/C3: If the classification system of the gross property consists (possibly near other static property classifications) only from T, or A, or C, or A and C, or T and A, or T and C-aspect property classification, or it does not contain one of these nor, then such property classification system incomplete. This conversely confutes the statement which is under (2), consequently the statement by (1) is true.(II.) If we again slightly reword this part of the theorem and use the introduced notations, we may write:
(1) For all complete S
impl is true that Nimpl≥2 and is therein T-A and T-C property classification. Let us suppose the opposite of this one:(2) For all complete S
impl is not true that Nimpl≥2 and is therein T-A and T-C property classification. Or other: There is such complete Simpl on which is true that: or (A) Nimpl<2 and/or (B) is not therein T-A and/or (C) not T-C property classification. We may write this statement (2) formally thus too:(3) $ S
implP(Simpl) where P(Simpl)=AÚ BÚ C and '$ ' is an existential quantifier (it means: 'there is such'), and the symbol 'Ú ' is the sign of the disjunction (aka: and/or); and P(Simpl)=AÚ BÚ C is on the Simpl relating statement (P= 'predicate'). [For example: logical value of AÚ B: AÚ B is true if, and only if, or only A, or only B, or A and B also is true. We may expand easily on the formula AÚ BÚ C the criterion of the truth, if we rewrite the formula AÚ BÚ C e.g. thus: AÚ BÚ C=(AÚ B)Ú C.]Since A and B and C denote, one by one, true or false statements (we may denote this thus too: 1='true'; 0='false'), hence possibly the number of truth values of AÚ BÚ C: 2
3=8. If we incorporate the following the table of truth values the meaning of A and B and C and their truth values, then we get the possibly truth values of AÚ BÚ C:
|
Table of |
|||||||||
|
truth-values |
|||||||||
|
|
A |
Ú |
B |
Ú |
C |
A |
B |
C |
A Ú B Ú C |
|
1. |
Nimp<2 |
|
T-A is not |
|
T-C is |
1 |
1 |
0 |
1 |
|
2. |
Nimp<2 |
|
T-A is |
|
T-C is not |
1 |
0 |
1 |
1 |
|
3. |
Nimp<2 |
|
T-A is not |
|
T-C is not |
1 |
1 |
1 |
1 |
|
4. |
Nimp≥2 |
|
T-A is |
|
T-C is not |
0 |
0 |
1 |
1 |
|
5. |
Nimp≥2 |
|
T-A is not |
|
T-C is |
0 |
1 |
0 |
1 |
|
6. |
Nimp≥2 |
|
T-A is not |
|
T-C is not |
0 |
1 |
1 |
1 |
|
7. |
Nimp<2 |
|
T-A is |
|
T-C is |
1 |
0 |
0 |
1 |
|
8. |
Nimp≥2 |
|
T-A is |
|
T-C is |
0 |
0 |
0 |
0 |
|
(1=true; 0=false) |
|||||||||
The 7th statement is formally true, but in fact it is self-contradiction. The 8th statement is formally a priori false, however verifies by content the part II./(1) of theorem. Hence we do not deal with these two statements.
Let us investigate so the first 6 statements.
We may diagnose on these that in the systems S
impl or is not T-A or is not T-C property classification, or is neither. But this one contradicts to the proved theorem T21/C2. The statement II./(2) leads to contradiction hence its opposite the statement II./(1) is true.We showed that the part I. and part II. of this theorem are true statements, consequently the theorem is true.
Q.e.d.
P.:
C.: T
21/C2, T21/C3Theorem 27: In the (0;M] interval the difference (or if the decrease is negative, algebraic sum) of same type measure data of the economic events, which occurred for increase and /or decrease of the gross property, is equal with the main sum of static classification of the gross property which belongs to the Mth time-point.
By axiom A5 in the property change class, in the (0;t] interval, apropos of the economic events came into existence difference (balance) of property increases and/or decreases equal to the static balance class in the tth time point belonging with sum, let it be either main or part sum.
Since by the condition t=M and the property change class is same to the gross property belonging with property change base class, hence, by
A5, in the (0;M] interval happened property changes' balance equal to with the main sum of static classification of the gross property in Mth time-point.Q.e.d.
P.: 1./
T28, 2./T5C.: 1./A
5.Theorem 28: With the formula A=C³ 0 represented incomplete property classification system (classic balance sheet) of gross property is transformable so that let it be complete.
By T
21/C3 with the formula(1) A=C³ 0
represented property classification system (classic balance sheet) of the gross property is incomplete, although is closed in point of the economist-specific economic events. If the formula (1) is transformable so that let it be satisfactorily informative, then by formula (1) represented the property classification system will be complete.
By definition: the property classification system is satisfactorily informative if it shows the economist's material position in a given time and at least the changes of its gross property to this time by the time-aspect property classification.
It is clear that with the formula E=F³ 0 represented property classification system shows the static material position in the t=M time-point. That let it be complete also it must show that the gross property is changing in (0;M] interval.
Therefore let us consider in tth time-point of (0;M] interval occurred all kth (k=1,2,...) economic event's same type measure data (by A
13), which is denoted with vk(t). (The sign of the decrease is negative.) The measure data of the only structuring events must be counted twice with opposite sign, or not at all. Namely the tth main sum is invariant in point of these (T19/C1). Let T denote the algebraic sum of the property changes vk(t). This time T is equal with the main sums A≥0 and C≥0 of the gross property by T27 in the Mth time-point, that is: (2) T=A=C≥0.Since these data v
k(t) are data of that economic events that occurred in (0;M] interval, where (0;M] divided on (t-1;t] parts (t=1,2,...,M), hence these data vk(t) change (by A5) the part sums I(t) of time classes of the (0;M] interval. Thus I(t) is equal with algebraic sum of the values vk(t) [by A5], that values vk(t) belong to the tth time-point. And T is equal with sum of all I(t) by A4. Hence
I(t)=T is true and (2) T=A=C≥0 also, which already clearly is a complete explicit N-fold (N=3) property classification system by T21.
Now let A
i(t) denote (i=1,2,...,n) the sum of changes of the ith asset-type in the (t-1;t] interval (t=1,2,...,M). That is: Ai(t) is equal algebraic sum of the values vk(t) [by A5], which values belong to Ai(t).Additionally let C
j(t) denote (j=1,2,...,p) the sum of changes of the jth capital-type in the (t-1;t] interval (t=1,2,...,M). That is: Cj(t) is equal algebraic sum of the values vk(t) [by A5], which values belong to Cj(t).Now if we sum the all A
i(t) by i and t, just as the all Cj(t) by j and t, then we obtain (by A4, A5 and T27) the following formula: PGR(M)=Ai(t)=
Cj(t)≥0. This formula already clearly is a complete implicit N-fold (N=2) property classification system by T21/C2.
Q.e.d.
P.:
C.: 1./A
4, A5, T19/C1, T21, T21/C2, T21/C3, T27.Thus we not only proved this theorem, but we showed that process also with which with the A=C≥0 formula represented property classification system will be complete. (This one can come in handy a shrewd for programmer, provided that perceives this information. J )
The law of the material position
GR(M)=Ai(t)=CE(t)+D(t)³
0 (where PGR(M)=Ai(t)D(t)³
0 and CE(t)0; and t=1,2,...,M; i=1,2,...,n) formula represents the law of the material position. Mean of this law: The man and all other economist, from its birth to its death, its existence in all tth moment, (1) has gross property [PGR(M)>0], bat then has debt also [D(t)>0], (2) D(t) in the good case is significantly less, in bad case is greater then the gross property, (3) or it does have neither its property [PGR(M)=0] its debt [D(t)=0] (this time it is pauper), (4) or its position worse at this one also because it has only debt [PGR(t)=0, D(t)>0] (this time it is pauper debtor). (5) And other case is not possible. (6) The material position of the economist and its all factors change in the time, the economist either economizes or leaves to itself its property, hence (7) its property, as its material position's either main factor, may investigate by N aspect (N≥3) , that is, at least by time, asset and capital-aspect (T29).
The following formula gives the model of the law of the material position:
PGR
(M)=Ai(t)=CE(t)+D(t)³
0 and PGR(M)=Ai(t)D(t)³
0, just as VS(t)0 (where t=1,2,...M; i=1,2,...,n).
Let us show the truth of statements (1),...(7).
(1) If the economist has property in tth time-point (that is P
GR(M)≥0 by T1), then it has debt also (by A7) and this time D(t)>0 by T2. The (1) statement so is true.(2) If the economist has property and thus debt also by (1), then on the relationship of these is true: P
GR(t)D(t)>0 by T1, T2 and T3. The (2) statement so is true.
(3) But there is such case that some economist does have neither property nor debt in a tth time-point (yet or already), that is: P
GR(M)=0 and D(t)=0 by A2, A7 (this time it is pauper). The (3) statement so is true.(4) But such tth time-point may be also, when the economist does not have gross property (that is: P
GR(M)=0 by A2, A7), however it has debt (so D(t)>0 by A7, T2 (this time it is pauper debtor). But this time its equity is negative [that is: if PGR(t)=CE(t)+D(t)=0 and D(t)>0 by A2 és A7, then D(t)=-CE(t)>0 by T5.] The (4) statement so is true.(5) And other case is not possible (by A
7). The (5) statement so is true.(6) The material position of the economist and its all factors change in the time, the economist either economizes (by T
20/C1) or leaves to itself its property (by T15/C) [the (6) statement so is true], hence(7) its property, as its material position's either main factor, may investigate by N aspect (N≥3) , that is, at least by time, asset and capital-aspect (by T
16/C3). The (7) statement so is true.For (1),...,(7) this theorem is true.
Q.e.d.
P.:
C.: A
2, A7, T1, T2, T3, T5, T15/C, T16/C3, T20/C1.
2. The base elements of theory of the property bookkeeping
2.1 Principles
2.11 Definitions of the property bookkeeping
2.111 The concepts of the general bookkeeping
(a1) A M=á S,X,Y,d ,l ñ
formula's symbols denoted system, as such mathematical (algebraic) model, which consists the ordered 5-tuple
á
S,X,Y,d
,l
ñ
.
From á S,X,Y,d ,l ñ the S, X and Y are such sets that have at least one element, that is: they are non-empty. The two symbols d and l are read on the product-set S´ X, that is: on the set of the ordered pair á s,xñ (where sÎ S,xÎ X), and these two symbols denote the d :S´ X® S and the l :S´ X® Y functions, where:
The functions d and l define the operation of the automaton A M by the following:
(a2) d (s,x)Î A
(a3) l (s,x)Î Y.
This automaton A M
is S-finite if status set S of A M is finite, and finite if the 3 sets of A M are finite.This automaton A M
is total defined if the functions d and l are defined on all ordered pair <s,x> (sÎ S,xÎ X), else it is called partial automaton.This automaton A M
is determined if the functions d and l are single-valued, else it is not determined.The automatons, which are investigated by me, are all total defined and determined A
M.We suppose yet the operation of these automatons A
M that they operate into discrete time-scale, that is, they can get input sign and give out the output sign only in discrete time-points[69]. Accordingly, if the automaton A M in some time-point is into the status sÎ S and this time it gets an input sign xÎ X, then it gets into the status d (s,x)Î S and it sends out an output sign l (s,x)Î Y (assuming that it is not the automaton without output sign).See the operation of the automaton A M
by the following schema (see below the figure a2):
On the Figure a2 the horizontal arrow [a,b] shows the status change of the automaton A
M by the input sign x (upper vertical arrow). This time the automaton came from a into status b. The lower vertical arrow signs the output sign outgiving.The temporary function of A
MA1:(a4) d (s,x)=s,
that is: the status of the automaton A
MA1 does not change if it get an input sign x, because S of A MA1 from x is independent.Clear that in case A
MA1 y depends only from x for (a4), hence the 's' may leave from output function l :(a5) l (s,x)=l (x)=y
.Hence the automaton A
MA1 may define following too:(a6) A
MA1=á S,X,Y,l ñwhere xÎ X; yÎ Y és ç Sç =1.
The (a6) algebraic model is reducible. We may leave everything from A
MA1 but the output function not. Namely, in point of fact the automaton A MA1 corresponds to the known formula y=¦ (x), where the set X is the domain of ¦ (x) and the set Y is the value-set of y. The difference only that X and Y both finite and countable, that is: the value of xÎ X and yÎ Y may be only discrete.Thus A
MA1 is equivalent with the following formula(a7) yi
=¦ (xi)wher i=1,2,...n and xiÎ
X; yiÎ Y.Additionally: in our case the depended variable of the model A
MA1 must be written with such vector to which are at least two elements[70]. This model is modifies:(a8) y
i=¦ (xi),where i=1,2,...n and y
i is a n-tuple (in the traditional bookkeeping this is 2-tuple, that is: ordered pair), xi contains the name (or the describe) of the standardized event, in point of fact it is a text-variable.Nevertheless the synthetical notation of the automaton A
MA1:(a8') A
MA1=á S,X,Y,l =yi=¦ (xi)ñwhere xiÎ X; yiÎ Y and yi is a n-tuple (i=1,2,...n) just as ç Sç =1.
2.112 The concepts of the property bookkeeping[73]
[yj'*,zj*,cj*,ei]j=[¦ j'(ei), g j(ei),cj*,ei]j
where:
The mean and value of the indexes:
This query automaton L
Q is symbolized with the next algebraic model (as ordered 7-tuple)L Q=á S,s0,P,R,Y,d ,l ñ
where:
The query automaton L Q has additional features:
The s0Î S is the automaton's beginner status-vector; its value always the null-vector (s0=0), in which L Q prompt returns if L Q passed on its mth status. The temporary function d of L Q yields this.
The status-vector sj=[sj1,sj2,sj3]*Î S is a 3-tuple of numbers. Its elements: the value of sj1 is to the j cumulated amount of the debit data (sj1=D); the value of sj2 is to the j cumulated amount of the credit data (sj2=C); the value of sj3 is to the j counted balance of D and C (sj3=B=D+C; where D³ 0 és C£ 0).
The already said bookkeeping data base R is the set of the bookkeeping data-records rj*=rj*(sj*,ei)Î R, and the number v of elements rj*: v=(l +q+u)+1 (for all j).
The row-vector yj*, as output word, is not other as the economic events of bookkeeping data base R with query (with filtration) obtained the row-vector yj* of the jth output data. Let this be with 9 elements. These are: yj*=(yj1,yj2,yj3,yj4,yj5,yj6,yj7,yj8,yj9)*, and yj* contains in order the data of the property-type p (in traditional bookkeeping this is the p account) by next: 1.) the identifier data p of the property-type (in traditional bookkeeping this is the identifier data p of the account); 2.) the time-point (t) of the economic events, 3.) the document's identifier data (d) 4.) the event's name (N), 5.) the sum of debit side (D³ 0), 6.) the total debit traffic (S D) 7.) the sum of credit side (C³ 0), 8.) the total credit traffic (S C) 9.) the sum of the balance: B=D+C. (Remark: in yj* or only D>0 or only C>0, both clearly are not possible positive because so real economic event does not exist.)
The temporary function of the query automaton L Q with two variables has the following formulas:
(a17) d (s0,p)=s1
(a17') d
(sj,rj*)
(j=1,2,...m).
The output function of the query automaton L Q with three variables has the following formula:
(a18) l (sj-1,rj-1*,p)=yj*.
The automaton L
Q as a matter of fact is a so-called sequential machine (see the figure a4 where we may see such graph which shows the operate of L
Q). If L
Q get the input sign p then it prompt moves from beginner status s0Î
S and takes up the status s1; then after this, the affect of these input signs r1*,r2*,...,rj*,...,rm* (rj*Î
R)
the L
Q gets into its middle states
s2,...sj,...sm-1Î S,
and it sends out the row-vector yj*Î Y of the output signs (j=1,2,...m-1). Finally, after the status sm the affect of input sign rm* it sends out the last output sign ym*Î Y then it returns into the beginner status s0Î S.
We may define the concrete form of the output function by the following:
Let us suppose that the bookkeeping data-record rj* has 6 elements. These are: rj*=[rj1=t,rj2=d,rj3=N,rj4,rj5,rj6], where is 1. the time-point (t) of the event, 2. the document number (d) of the event, 3. the name (N) of the event, 4. the identifier (rj4) of the debit property-type (debit account in the traditional bookkeeping), 5. the identifier (rj5) of the credit property-type (credit account in the traditional bookkeeping), 6. money sum (rj6) of the property change.
Additionally let Idp be an indicator function, its value 1 if p same the value of the debit identifier xj4 else 0; and let Icp be an other indicator function, its value -1 if p same the value of the credit identifier xj4 else 0.
Let D denote the debit sum of the property-type p and let D=Idp× xj6, and let C denote the credit sum of the property-type p and let C=Icp× xj6, (j=1,2,...m).
Let S D denote the sum of the debit traffic and let
S D=sj1=sj-1,1+D (j=1,2,...m).
Let S C denote the sum of the credit traffic and let
S C=sj2=sj-1,2+C (j=1,2,...m).
Yet let B denote the value of the status variable sj3 as balance-memory (j=1,2,...m), and thus
B=sj3=sj-1,3+(D+C) (j=1,2,...m).
The values of elements of the output-vector yj*
(a22) yj*=[p,t,d,N,D,S D,C,S C,B] (j=1,2,...m).
The figure a4 shows this initial and modified Mealy automaton L Q the operating schema with directed graph below (Figure a4):
L
Is ordered 9-tuple, where
- P
={ p1,p2,...,pk,... pq} is the set of q pieces class identifiers of the A or C-aspect all class (they are accounts in traditional bookkeeping) of the property (P is the input-sign set 1.);- R is the bookkeeping database (R is the input-sign set 2.);
- Y={ l (sm,rm*)=y*} is the set of L TB which consists an element the only output-vector, where sm is the final status of L TB;
- a p(s0,pk+1)=pk (k=0,1,2,...,q-1) is that input function which defines the value of the next input sign pk if L FK gets into s0;
- d is the temporary function (cf. definition 5);
- l is the output function (cf. definition 5);
- F={ sm} the set of final status of the L TB it consists only element sm.
The figure a5 below shows the operating of L FK:
2.12 The axioms of the property bookkeeping
2.121 The documentary principle
2.122 The general bookkeeping principles of undefinable imperishableness of the truth-untruth dilemma
P.: 2./T
3.
P.: 2./T
4.
P.: 2./T
4.2.123 The principle of the inadequate controller automatons
P.: 2./T
4.2.124 The principle of economist-dependent of the abstract events
P.: 2./T
5.2.2 The theorems and their proofs
1).Let e
i denote the data-vector of the ith economic event (i=1,2,...,n). But we may book only from document's data (by 2./A1), hence let di denote the data-vector of the ith documented economic event and let bi denote the data-vector of the ith bookkeeping event (see the definitions). The vector di differs from vector ei in point of data's intension only in its document identifier data a2(di) by relating definitions and the following figure e1. But there is not need on the document identifier data in point of characterization of change of the material position. Hence we can abstract from this data, as if it would not also be. Consequently it is true in point of characterization of change of the material position that eiº di (the symbol 'º ' means e.g. that 'ei and di is equivalent').However b
i differs from ei and di only in the data a4(bi) of the event coordinates. This data a4(bi) is not other as the data a3 of di and bi or rather the data a2 of ei which is the formalized version of the verbal data 'name (or describe) of event', that is, it is the data of the event coordinates by definition. We may say that both the verbal a2(ei), a3(di) and a3(bi) just as the formalized a4(bi) denote the final property classes and their change's character (such as increase or decrease or the structure's change). Consequently this data are equivalent in point of their contents (1./A14). [Actually the data a3(bi) is in bi only for control and verbal notation (name).] However we can abstract from data a2(bi) the already said for cause, as if it would not also be. Thus we get with the comparison of content of the data-structures the followings:The data's content e
i di di bi[time-point] a
1(ei)º a1(di) and a1(di)º a1(bi),[event-name] a
2(ei)º a3(di) and a3(di)º a3(bi)º a4(bi),[quantity] a
3(ei)º a5(di) and a4(di)º a5(bi),[money-value] a
4(ei)º a6(di) and a5(di)º a6(bi),Figure e1
That is: from the data-content which influences the material position the followings hold: eiº d
i and diº bi . But then is true that: eiº bi because the equivalence is transitive.Q.e.d.
, .C.: 1./A14, 2./A
1. 2).Let E be a finite and non-empty set, wherein the economic events eiÎ E (i=1,2,...,m), in a given interval, change the main sum of property and the debt just as upbuild their property classification system by 1./T20/C7. Additionally let d
i denote a documented economic event and let D be a finite and non-empty set of the documented economic events, that is: diÎ D. And let B denote an also finite and non-empty set of the bookkeeping events which perform in the bookkeeping registration B, that is: biÎ B.Now let us assign by a given rule the economic events to the bookkeeping events, but the data of the economic events also contains document (by 2./A2) or rather documented economic event interposing. This operate, also known as mapping, so will be indirect. (It also is called to composite mapping.)
1) The first object-element is the economic events ei.
Let j be a mapping from E onto D. The j means that the mapping's rule is following: the data of the documented economic event diÎ D is same the economic event eiÎ E with by content corresponding data (i=1,2,...,m), aside from the identifier of the document.
2) The first image-element is di that is the documented economic event. This mapping j is mutually unequivocal, because to any economic event ei belongs one and only one documented economic event di and vice versa (by 2./T1). We can express this mapping j with the following symbols too: ei« j (ei)=di and the symbol '« ' means that the mapping j is mutually unequivocal. This image-element di is at the same time the second object-element together with its data.
Let y be a mapping from D onto B. The y means that the mapping's rule is following: the data of the documented economic event diÎ D is same the bookkeeping event biÎ B with by content corresponding data (i=1,2,...,m) (by 2./T1).
3) The second image-element is bi that is the bookkeeping event. We can express this mapping y with the following symbols too: di« y (di)=bi; thus y is mutually unequivocal.
The composite mapping is the following:
ei« y [j (ei)]=bi (i=1,2,...m).
Additionally: the algebraic sum of quantity or monetary value data of the economic events eiÎ E (i=1,2,...,m), in a given interval, change and upbuild the main sum of property and the debt just as their property classification system by 1./T20/C7.
But eiº bi by 2./T1, thus the theorem is true.
Q.e.d.
P.: 2./T2/C1, C2.
C.: 1./T20/C7, 2./A1, T1.
Corollary 1: The image of the bookkeeping registration as image of factors and changes of the material position and this mapping's object in point of its character is necessarily equivalent (2./T2/C1).
Q.e.d.
P.:
C.: 2./T2.
Corollary 2: The theorems and laws of the property theory hold in same form and with same content in the bookkeeping too (vice versa in generally this is not true) because in the property theory given system and the bookkeeping's system are isomorphic (2./T2/C2).
Q.e.d.
P.:
C.: 2./T2
The theorem otherwise:
The probability (p) of unfaulty imperishableness of the uncontrolled property bookkeeping is always greater then zero and less then one at a given time-point t (0<p<1).
Let EF
denote the event that the uncontrolled property bookkeeping is faulty, EUF that it is unfaulty. Let H denote that EF and EUF occurred together, this time we write: EFÈ EUF=H. H means that in point of the faulty or unfaulty imperishableness of the uncontrolled property bookkeeping subsequential total set of the events or other: total event-system of it. EF and EUF close off each other pair-wise [EFÇ EUF=
]; that is: together both can not occur. However then some but alone either of them occurs certainly. Additionally let 
denote the complement of H on which the following formula holds:
.
The expressive denotation of is that: E
' denotes this).
The probability of set H (that is: the probability of the certain event) is 1 by the axiom II of the probability theory. We can denote this with P(H)=1. Because that event occurs certainly that: the uncontrolled property bookkeeping registration will be faulty or unfaulty in a given time-point t. Let P(EF
)=p' denotes the measure of probability of faulty imperishableness of the uncontrolled property bookkeeping and P(EUF)=p denotes the measure of probability of faulty imperishableness of it.This time I have: 0<p<1.
Now by 2./A2
: "The probability (p') of faulty imperishableness of the uncontrolled property bookkeeping is always greater then zero and less then one at a given time (0<p'<1)."But the uncontrolled property bookkeeping registration has two possible outputs which are close off each other [EFÇ
EUF=] and they create a total event-system [EFÈ
EUF=H]. Hence the following formula holds by the axioms[74] I, II and III of the probability theory:
(a) 0<P(EFÈ EUF)=P(EF)+P(EUF)=P(H)=1.
The following inequality (b) arises from equality P(E
F)=p' and P(EUF)=p with substitution p' and p into (a):(b) 0<p'+p=1.
But we get from equality p'+p=1
(c) p'=1-p.
However 1>p'>0 holds by 2./A
2. This time we may use again that p'=1-p. We may write for p'(d) 1>1-p>0.
But we can get with rearrangement from formula (d): 0<p<1. So this formula (d), as the statement of this theorem, is true, where 1 equal with the 100 percent of the immaculacy.
Q.e.d.
P.:
C.: 2./A
2.Theorem 4: The uncontrolled inventory (that is: if it is unchecked with the documents of the corresponding economic events) does not prove the unfaulty of the uncontrolled bookkeeping[75] and with its data made balance sheet (2./T
4).Let us have the opposite of the theorem:
T4'
: The uncontrolled inventory (that is: if it is unchecked with the documents of the corresponding economic events) prove the unfaulty of the uncontrolled bookkeeping and with its data made balance sheet.1) Let us suppose that 10 pieces barrows, as current assets, perform, near many other, in a builder's bookkeeping registration and by it compiled balance sheet in 10x monetary value. But at stock-taking the employed of builder found 11 pieces barrows clearly in 11x monetary value. However, nobody did not check the data of the inventory, that is nobody did not compare with the data of the base documents the data of inventory, but the bookkeeping's data nor. That is e.g. nobody did not compare the data of the barrow-shopping documents with the data of inventory and bookkeeping.
The following question arises: Which data is real certainly? Is the data of the bookkeeping and by it compiled balance sheet or the corresponding data of the inventory real?
We can not prove nor confute the authenticity of the bookkeeping and balance sheet by the formulas A=C and/or å Dt=å Cr because both are not able on the target of check (by 2./A5
).That dilemma without checking is undefinable that the data of the bookkeeping and the balance sheet are unfaulty or not (by 2./A3
). For example: the data of bookkeeping or the balance sheet can unreal if the accountant misprizes the number of pieces at bookkeeping, or if the booked invoice's quantity and monetary value data for the data of the thing's takeover justifying parcel bill is mistaken; and the assign did not observe this ones for did not checked them.But it is possible that the data of the bookkeeping and the balance sheet are real however the corresponding data of the inventory are faulty. For example: if the employed, who takes stock, badly sums the numbers of the barrows, and writes 11 pieces for 10 and so the 11 was multiplied by x. Or the employed wrote for inattention the 11 for 10. But it is also possible that, if there were not inventorial identifiers on the barrows, on the seat of builder firm the lonely barrow of the fence-repairman also was added to the 10. So: the unfaulty of the inventory data, without their checking, neither may not prove nor may not confute. This problem without checking is undefinable (by 2./A
4).However these facts already contradict to T4
' hence this theorem (2./T4) is true.2) We may use also these arguments then if both the bookkeeping with the balance sheet together and the inventory contain 10 barrows and its 10x monetary value, but neither the bookkeeping nor the inventory are not controlled, that is: both are not fulcrate with base documents. Only now we do not know that the conformity really holds or not, this one holds for same cause as in the case 1). So: neither may not prove nor may not confute the conformity. This problem without checking is undefinable (2./A3
, 2./A4). This fact also contradict to T4' hence this theorem (2./T4) is true.That is: the corresponding data of the uncontrolled bookkeeping/balance sheet and the uncontrolled inventory either differ [in case 1)] or square [in case 2)] the uncontrolled inventory neither does not prove nor does not confute the differing or the conformity.
Q.e.d.
P.: 2./T4
/C1, C2.C.: 2./A
3, A4, A5.Corollary 1: The uncontrolled (that is: with the document of the corresponding economic events and with the corresponding data of the controlled inventory unchecked) bookkeepeng events (account items) do not confirm (that is do not prove) the bookkeeping registration and by its data made balance sheet authenticity (2./T
4/C1).P.:
C.: 2./A
4. i, Rd, Rb, Re denote on the straight the reality of on the same season relating inventory data, all document's data, all bookkeeping data and the all economic event's data. By itself neither the inventory's data (Ri) nor to the inventory season corresponding data (Rd) of the booked documents but yet the both together nor confirm the reality of the touched bookkeeping (Rb) and balance sheet but only the following four equality all at once: Re=Rd and Rd=Rb and Rb=Ri and Re=Ri (2./T4/C2). This corollary is called the law of "square control" of the bookkeeping.P.:
C.: 2./A
4.Standardization and automatization
Theorem 5: We can squarely assign a such finite set to any economist which consists its activity belonging from the standardized economic events (2./T
5).Let
E denote some economist, '® ' the inherence and Ea the set of such abstract economic events which's elements are specific for the activity of the economist E and arose with the abstraction of the bookkeeping events. Let ea denote these abstract economic events, hence eaÎ Ea is holds. However let es denote the standardized economic events and Es their finite set, that is: esÎ Es. This time we can write this theorem with these notations thus too: (1) E® Es. Let us show that (1) is true.By the axiom 2./A6
: We can squarely assign a such finite set to any economist which consists from pair-wise different abstract economic events Ea and this events are able on it that we define with them the possible changes of material position of the economist. But if we can squarely assign a so finite set to any economist then we can assign squarely to E also, that is we may write: (2) E® Ea.However, by the relating definition: I name to standardized economic events (briefly only: to standardized event) those event-names that in the economist's bookkeeping required to use the concrete bookkeeping events on strictly name, by data-type a3. We constitute these events with standardization from the abstract economic events; and they all differ one from another formally (in view of the words or its the name's order or in view of the event's description) and by content (in view of the mean of the name or description of the event) too, additionally their event coordinate n-tuple is clever, that is, real. But es=eaÎ Ea for this definition, hence EsÍ Ea
holds and then E® Es is true.Q.e.d.
P.: 2./T5
/C1, C2, 2./T7.C.: 2./A
5.Corollary 1: Let n denote the number of the abstract economic events ans k the number of the standardized economic events. On the relation of these holds: 1£ k£ n (n=1,2,...) [2./T5/C1].
P.: 2./T
6, 2./T7.C.: 2./T5.
Corollary 2: The standardized economic events are also typical on the economist's activity, that is: they are economist-specific [2./T5/C2].
P.:
C.: 2./T5.
Theorem 6: In the (0;t] interval (t=1,2,...,M) let concrete bookkeeping events occur that are named with standardized economic events. Let us classify these bookkeeping events by types of these standardized economic events. Thereafter let us sum these gross property changes that are occurred apropos of these bookkeeping events. This time the got algebraic sum, which belongs to the Mth time-point, is equal with the main sum of classification (by types of these standardized economic events) of the gross property changes (2./T
6).In the (0;M] interval let them occur n pieces (n=1,2,...) economist-specific economic events. Thus all time the measure of the gross property increases or decreases or only changes its structure (by A15). Let V denote the measure of the gross property. Let us suppose that the measure data of the economist-specific bookkeeping event shows with the negative sign the decreases. Additionally, since the main sum V of any classification of the gross property is invariant in point of compensatory (the opposite sign but equal in size) changes of its two part sums (by 1./T19/C1), hence we can regard to zero the measure data of the compensatory changes.
With these conditions the measure data V of the gross property in Mth time-point is equal with the algebraic sum of same type measure data of in the (0;M] interval recorded n pieces economist-specific bookkeeping events (by 1./T
27). But we can name these n pieces economist-specific bookkeeping events with its corresponding standardized economic events, however hereof V clearly is not changing.The number of these standardized economic events is k, where 1≤k≤n (n=1,2,...) [by 2./T
5/C1].Now, if k=n that is if the number of the standardized economic events and the number of the bookkeeping events are identical, this theorem is clearly true.
However if 1<k<n then let us decompose the summation onto k pieces of group by the type of the standardized economic events. Finally let us sum the algebraic sums of these groups, but this operate does not modify the value of V for the associativity of the summation (by 2./T
6).Q.e.d.
P.:
C.: 1./T19/C1, T27,
A15, 2./T5/C1.Theorem 7: We can assign squarely to all standardized economic event the to it corresponding to concrete bookkeeping event belonging event coordinates' data-vector y'*=o* as meta data of the class-coherence [in the traditional bookkeeping it is called: to debit-credit account coherence] (2./T7).
Let y'*=o* denote the data-vector of some bookkeeping event which is same with the data a4
of this bookkeeping event.The economic events' to changed part sum, within time classes, belonging final property classes and its change character marked data (that is the event-name or event-description) and the data-vector of the event coordinates mutually and squarely correspond to one another by their meaning (for A14). From this one and from the theorem 2./T
1 it follows that all standardized economic event which is used to name some concrete bookkeeping event coordinates' data-vector y'*=o* as meta data of the class-coherenceCorollary: To all standardized economic event may be squarely to assign all such concrete data of to them corresponding concrete documented economic event what depend from these standardized economic event (2./T7/C).
Q.e.d.
P.:
C.: 2./T
7.Theorem 8: The coordinates of some bookkeeping event of the economist may define automatically like the function of the economist-specific standardized economic events. (2./T8).
Let ei denote the ith economist-specific [2./T5/C2] standardised economic events (i=1,2,...,n). We can assign squarely to all standardized economic event the to it corresponding to concrete bookkeeping event belonging event coordinates' data-vector y'*=o* as meta data of the class-coherence [in the traditional bookkeeping it is called: to debit-credit account coherence] (2./T7). But y'*=o* is the value of the bookkeeping derivative[76]
yi'*=
Now the Mealy's event-coordinate designator automaton
yi'*=[y1,y2,...yk]i'=oi*=j
'(xi)=¦
'(ei) automatically defines the y'*=o* as the value of the bookkeeping derivative in the position xi
Q.e.d.
P.: 2./T
8/C.C.: 2./T
7, 2./T5/C2.Corollary: If the ei® yi'*=[y1,y2,...yk]i
'=oi* is all for i correctly predefined then the class-coherent of any and any number of documented economic or rather bookkeeping events will be unfaulty when we give with event-coordinate designator automaton. That is if all event-coordinate designation unfaulty for all ei then the use of the event-coordinate designator automaton excepts the faults of the event-coordinate designation. This one isolate the bookkeeping system from this type of the fault, for any ei and all the same that how many times we repeat this operate. (2./T8/C).P.:
C.: 2./T
8.Theorem 9: The data of some bookkeeping event of the economist may define automatically like the function of data pf the economist-specific standardized economic events and the concrete documented economic events with bookkeeping automaton (2./T9).
The bookkeeping automaton is such complex real automaton which is modeled with more different simple Mealy's automaton. This one give automatically the class coherence data in function of standardized economic events (2./T
7) just as all such data which depends from standardized economic event ei (i=1,2,...,n) (2./T7/C). It may be so data as for example the measure-unit or the per cent of the value added tax (VAT) or the VAT's suggested[77] sum, etc. The output data of the bookkeeping automaton may be yet the concrete data of the concrete documented economic events too. For example these are the date of event, the identification of the document, the quantity, the monetary value, etc. These will be output-data also in one block.Q.e.d.
P.: 2./T
10C.: 2./T
7, 2./T7/C.Theorem10: The data of the (trial balance) cumulative statement what as the data of the A or C-aspect and the identification sign siÎ S={ s1,s2,...,si,...,sp} property's type (traditionally these called to ledger's accounts) may define
with cumulative statement maker (trial balance query) automaton made from the data-base (2./T10).The A and C-aspect classifications and the all identification sign siÎ S={ s1,s2,...,si,...,sp} property's type (traditionally these called to accounts of the ledger) and the part sums of these as the data of a (trial balance) cumulative statement can query automatically with cumulative statement maker (trial balance query) automaton from the bookkeeping data-base, which made with bookkeeping automaton by 2./T9
. Namely the real version of the cumulative statement maker (trial balance query) abstract automaton is a such sequential machine which in sequent define by the input signs s1,s2,...,si,...,sp marked traffic and balance data of the property's type (traditionally these are the ledger's accounts) just as it orders these in a cumulative statement (in a trial balance).Q.e.d.
P.: 2./T10
/C1,C.: 2./T
9.Corollary 1: We may make the balance sheet also with the cumulative statement maker (trial balance query) automaton from the bookkeeping database if we correspondently complete this automaton (2./T10/C1)
.P.:
C.: 2./T
10. .P.:
C.: 2./T
10.Corollary 3: For the use of the bookkeeping and the cumulative statement maker (trial balance query) automaton is unnecessary to make and to keep with computer or hand the traditionally ledger's accounts. Ergo the account theories in this situation are reason lost. This situation is the death of the account theories (2./T10/C3).
P.:
C.: 2./T
10.
3. The base elements of theory[78] of the debit and circle-debit
P.: 3./T2
, T3.
P.: 3./T2
3.2 The theorems of the debit and circle-debit and its proofs
Theorem 1: All creditors are simultaneously debtors too (3./T
1).Let G1 be one of the existent creditors and let it credit e.g. to the economist G0. Let us show that like G1 so all creditors are simultaneously debtors too.
If G1 is creditor of G0 then G0 owes to G1 by 1./A
9 and hence G1 is a man of property by 1./A8. (Let V1 denote the property of G1, where V1>0 by 1./T1.) But then G1 has debt also because whom has property it has debt also by 1./A8. (Let A1 denote the debt of G1, where A1>0 by 1./T2.) Thus the next statement hold true: (V1>0)® (A1>0) by 1./A8 and 1./T1,T2. Consequently it must be that let G1 owes one or more to somebody that is either to G0 or other to economist(s) e.g. to G2. Ergo G1 is debtor itself too. Since G1 may be any one of creditors (that it is debtor it follow thereout that it in one block creditor by 1./A8) hence the following statement is true: all creditors are simultaneously debtors too.Q.e.d.
P.: 3./T3.
C.: 1./A
8,A9,T1,T2.Theorem 2: The performers of the market are all economists of property (3./T
2).Let us choose out arbitrary two of the performers of the market. Let them be G
1 and G2. This time for example let G1 seller whilst G2 buyer by 3./A1. I have: G1 and G2 are economists of property as either of others.If namely not then: or G1
as seller do not sells away the own property's object or G2 as buyer do not give across as the quid pro quo of the own property's object or neither of them give the own property's object in the transaction to the other one. But these three case contradict to 3./A2 because by 3./A2: "The performers of the market apropos of the traffic always sell away the object of the own property as seller or rather they give that one as quid pro quo if they are buyers." Thus G1 and G2 propertied may be only. If however G2 is the seller and G1 is the buyer, they are then also propertied performers of the market sithence this time we transpose only the indexes of Gs.Finally: since we may choose of the all market performers arbitrary two to G
1 and G2 hence what statement is true in case of G1 and G2 that is true in case of the all market performers too.Q.e.d.
P.: 3./T3
, T4.C.: 3./A
1, A2.Corollary: All seller is buyer also and vice versa (3./T2/C)
.P.:
C.: 3./T
2.Theorem 3: If there are only two economists of property on a market then they owe only to one another. This time they, as debtor-pair, make that debtor-circle to which is minimal the number of its members. This one is the minimal case of the circle-debit. (3./T
3).There are at least two performer on all market (by 3./A
1) and they are economists of the property (by 3./T2). Thus let two performers be only now on the market; they are G1 and G2. Let us show that G1 and G2 owe only to one another.Since G1
has property hence G1 has debt (by 1.A1) and thus G1 has creditor too (for 1./A9). This creditor can not be only G2 by the condition. But if G2 is creditor then G2 has property (by 1./A9) and hence G2 has debt (by 1.A1) just as thus G2 has creditor too (for 1./A9). However this creditor can not be only G1 by the condition.G1
and G2 thus owe only to one another. This one is the minimal case of the circle-debit (3./T3).Q.e.d.
P.: 3./T
4.C.: 1./A
8, A9; 3./A1, T1, T2.Theorem 4: There is circle-debit on all market that is: the circle-debit is the attribute of the markets other their essential feature (3./T
4).Let n denote the number of the market performers. Let there be n market performers (n³ 2) on the examined market. I have that there is circle-debit on this market and on all markets, consequently the circle-debit is natural and essential feature that is attribute of the markets. Hence we must show that there is at least a debtor-circle on the n-performers market (n³ 2).
There is n performers (n³ 2) on this examined market. These are all economists of property.
If n=2 that is there is only two performer of this market then these two as debtor-pair make a minimal member debtor-circle. This time the theorem is true. Whereas this market is not specified in other interrelation hence it also is true statement that there is circle-debit on all two-performer market.
Now let us show that there is on this market debtor-circle that is circle-debit if n>2. We can apply to this the following method too:
All this denotes that there is at least a debtor-circle on this market to which has n or n-1 or ... 3 or 2 members.
Whereas this market is not specified in other interrelation hence it is true statement that there is circle-debit on all market and this is the attribute of the markets (3./T
4).Q.e.d.
P.:
C.: 1./A8
, A9; 3./T2, T3.Q.e.d.
P.:
C.: 3./T
4. max of the debtor-pairs Pmax=[(n-1)*n]/2 what is equivalent e.g. with number of that lines what connect apexes of a convex n-angle (where n³ 3). Pmax=[(n-1)*n]/2 may verify easy with mathematical induction. [3./T4/C2].Q.e.d.
P.:
C.: 3./T
4.Q.e.d.
P.:
C.: 3./T
4.Q.e.d.
P.:
C.: 3./T
4.
Though main theme what we discuss in this book is the bookkeeping of the property, still I recommend the following theme, which follows from theory system in this book written, in the note of readership:
The individual bookkeepings may have differential objects. For example: the losing team keeps the books its thrashing; the football-umpire keeps the books the distributed yellow and red cards; the pedagogue books the grade of student; the historian books the events of the history; the biologist books the results of the research; the economist's accountant in turn books the data of the changes, that is the data of the economic events, what occurred in the property and/or debit apropos of the economy. Thos ones thus are all events of chronologic registrations that is bookkeepings namely special bookkeepings.
Special bookkeepings are thus for example:
I will show three well illustrating examples on the special bookkeepings in the following three appendixes, whilst this ones are fictive.
Bookkeeping of property and its balance sheet
The table y1 below shows a n=4 that is time-assets and time-capitals aspect dynamic just as assets and capitals aspect static complex balance sheet of the property which is defined squarely in the table y2 visible by bookkeeping database of the property.
|
4 aspect complex dynamic and static balance sheet | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
|
Time-assets dynamic classification |
Static classification | ||||||||||
|
1 |
2 |
3 |
|
|
|
4 |
|
|
|
|
|
5 |
|
Assets-types |
|
Times (days) |
a(i)s |
|||||||||
|
|
a |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
|
Sheep |
1 |
1 000 |
10 |
-10 |
|
|
|
|
|
|
|
1 000 |
|
Cow |
2 |
2 000 |
|
|
|
|
|
|
|
|
-1 000 |
1 000 |
|
Feed |
3 |
|
|
|
|
|
|
|
|
600 |
|
600 |
|
Claims from buyers |
4 |
300 |
|
|
-200 |
|
|
|
|
|
700 |
800 |
|
Gold money |
5 |
|
|
|
200 |
|
50 |
150 |
1 100 |
-600 |
700 |
1 600 |
|
p(t) |
|
3 300 |
10 |
-10 |
0 |
0 |
50 |
150 |
1 100 |
0 |
400 |
5 000 |
|
Cumulated p(t) |
|
3 300 |
3 310 |
3 300 |
3 300 |
3 300 |
3 350 |
3 500 |
4 600 |
4 600 |
5 000 |
|
|
|
||||||||||||
|
|
Time-capitals dynamic classification |
Static classification | ||||||||||
|
1 |
2 |
3 |
|
|
|
4 |
|
|
|
|
|
5 |
|
Capitals-types |
|
Times (days) |
c(j)s |
|||||||||
|
|
c |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
|
Cumulative result |
1 |
2 700 |
|
|
|
|
|
|
|
|
|
2 700 |
|
Services income |
2 |
|
|
|
|
|
50 |
150 |
|
|
|
200 |
|
Revenue |
3 |
|
|
|
|
|
|
|
|
|
1 400 |
1 400 |
|
Wagework revenue |
4 |
|
|
|
|
|
|
|
1 100 |
|
|
1 100 |
|
Accretion |
5 |
|
10 |
|
|
|
|
|
|
|
|
10 |
|
Labour cost |
6 |
|
|
|
|
|
|
|
|
|
-100 |
-100 |
|
Goods cost |
7 |
|
|
|
|
|
|
|
|
|
-1 000 |
-1 000 |
|
Long-term liabilities |
8 |
500 |
|
|
|
50 |
|
|
|
|
|
550 |
|
Current liabilities |
9 |
100 |
|
-10 |
|
-50 |
|
|
|
|
100 |
140 |
|
Opener balance |
10 |
|
|
|
|
|
|
|
|
|
|
0 |
|
p(t) |
|
3 300 |
10 |
-10 |
0 |
0 |
50 |
150 |
1 100 |
0 |
400 |
5 000 |
|
table y1 |
||||||||||||
|
Bookkeeping registration (database) | |||||||
|---|---|---|---|---|---|---|---|
|
1 |
2 |
3 |
4 |
5 |
|||
|
Time- |
The name (description) of bookkeeping event |
Event- |
Sum |
||||
|
point |
|
|
|
|
coordinates |
(gold) |
|
|
(Day) |
Opening of assets: |
|
|
(D) |
(Cr) |
|
|
|
0. |
1. Opening of sheep-stores |
|
1a |
10c |
1 000 |
||
|
0. |
2. Opening of cow-stores |
|
2a |
10c |
2 000 |
||
|
0. |
3. Opening of cleims |
|
4a |
10c |
300 |
||
|
|
Opening of capitals: |
|
|
|
|
||
|
0. |
4. Opening of cumulated result |
|
10c |
1c |
2 700 |
0. |
5. Opening of long-term liabelities |
|
10c |
8c |
500 |
|
0. |
6. Opening of current liabilities |
|
10c |
9c |
100 |
||
|
|
Current economic events: |
|
|
|
|||
|
1. |
7. Account of the sheep-accretion |
|
1a |
5c |
10 |
||
|
2. |
8. Extinction of the sheep-debt in nature |
|
-1a |
-9c |
10 |
||
|
3. |
9. Payment of claim in cash (gold money) |
|
5a |
-4a |
200 |
||
|
4. |
10. Across class of the current liability on long-term |
8c |
-9c |
50 |
|||
|
5. |
11. Service income in cash |
|
5a |
2c |
50 |
||
|
6. |
12. Service income in cash |
|
5a |
2c |
150 |
||
|
7. |
13. Wagework revenue in cash |
|
5a |
4c |
1100 |
||
|
8. |
14. Shopping of the feed in cash |
|
3a |
-5a |
600 |
||
|
9. |
15. Sale income of one cow for cash |
|
5a |
3c |
700 |
||
|
9. |
16. Cost of two sold cows |
|
-2a |
-7c |
1000 |
||
|
9. |
17. Sale income of one cow in credit |
|
4a |
3c |
700 |
||
|
9. |
18 Account of payment of labour as cost and as liability |
9c |
-6c |
100 |
|||
|
table y2 |
|||||||
|
TIME-ASSETS-ASPECT CUMULATIVE CLASSIFICCTION |
ASSETS STATIC CLF. | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Asset-types |
|
Time-points (days) |
ĺ a(i) |
|||||||||
|
|
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
|
Sheep |
|
1 000 |
1 010 |
1 000 |
1 000 |
1 000 |
1 000 |
1 000 |
1 000 |
1 000 |
1 000 |
1 000 |
|
Cow |
|
2 000 |
2 000 |
2 000 |
2 000 |
2 000 |
2 000 |
2 000 |
2 000 |
2 000 |
1 000 |
1 000 |
|
Feed |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
600 |
600 |
600 |
|
Claims from buyers |
|
300 |
300 |
300 |
100 |
100 |
100 |
100 |
100 |
100 |
800 |
800 |
|
Gold money |
|
0 |
0 |
0 |
200 |
200 |
250 |
400 |
1 500 |
900 |
1 600 |
1 600 |
|
total property |
|
3 300 |
3 310 |
3 300 |
3 300 |
3 300 |
3 350 |
3 500 |
4 600 |
4 600 |
5 000 |
5 000 |
|
|
||||||||||||
|
TIME-CAPITALS-ASPECT CUMULATIVE CLASSIFICATION |
CAPITALS STATIC CLF. | |||||||||||
|
Capita-types |
|
Time-points (days) |
ĺ c(j) |
|||||||||
|
|
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
|
Cumulative result |
|
2 700 |
2 700 |
2 700 |
2 700 |
2 700 |
2 700 |
2 700 |
2 700 |
2 700 |
2 700 |
2 700 |
|
Services income |
|
0 |
0 |
0 |
0 |
0 |
50 |
200 |
200 |
200 |
200 |
200 |
|
Revenue |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 400 |
1 400 |
|
Wagework revenue |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 100 |
1 100 |
1 100 |
1 100 |
|
Accretion |
|
0 |
10 |
10 |
10 |
10 |
10 |
10 |
10 |
10 |
10 |
10 |
|
Labour cost |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-100 |
-100 |
|
Goods cost |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-1 000 |
-1 000 |
|
Long-term liabilities |
|
500 |
500 |
500 |
500 |
550 |
550 |
550 |
550 |
550 |
550 |
550 |
|
Current liabilities |
|
100 |
100 |
90 |
90 |
40 |
40 |
40 |
40 |
40 |
140 |
140 |
|
Opener balance |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
total property |
|
3 300 |
3 310 |
3 300 |
3 300 |
3 300 |
3 350 |
3 500 |
4 600 |
4 600 |
5 000 |
5 000 |
|
table y3 |
||||||||||||
Time-curve balance sheet to the table y3
Bookkeeping of level of knowledge of the students and the knowledge balance sheet
The data of data-base of the bookkeeping define (see in the table t4) the balance sheets in the table t1, t2 and t3. A composite static and dynamic balance sheet version below is visible in the table t1. The balance sheet by t3 contains the known school report of the student in the first two columns.
|
STATIC and DYNAMIC (composite) KNOWLEDGE BALANCE SHEET | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
|
|
Students |
Average knowledge- level |
|
Subjects |
Average knowledge- level |
|
Months |
Average knowledge- level |
||
|
1 |
Benedict Jacob |
3 |
1 |
Mathematics |
2 |
1 |
January |
5 |
||
|
2 |
Peter Smith |
3,5 |
2 |
Physics |
3 |
2 |
February |
4 |
||
|
3 |
Mary Gross |
4 |
3 |
Literature |
4 |
3 |
March |
2,5 |
||
|
|
|
|
4 |
History |
5 |
|
|
|
||
|
|
|
3,5 |
|
|
3,5 |
|
|
3,5 |
||
|
Table t1 |
||||||||||
|
STATIC KNOWLEDGE BALANCE SHEET | ||||||||||
|
|
Students |
Average knowledge- level |
|
Subjects |
Average knowledge- level |
|||||
|
1 |
Benedict Jacob |
3 |
1 |
Mathematics |
2 |
|||||
|
2 |
Peter Smith |
3,5 |
2 |
Physics |
3 |
|||||
|
3 |
Mary Gross |
4 |
3 |
Literature |
4 |
|||||
|
|
|
|
4 |
History |
5 |
|||||
|
|
|
3,5 |
|
|
3,5 |
|||||
|
Table t2 |
||||||||||
|
STATIC STUDENT- SUBJECT KNOWLEDGE BALANCE SHEET | ||||||||||
|
|
Students |
Average knowledge- level |
|
Subject of students |
Average knowledge- level |
|
Subjects |
Average knowledge- level |
||
|
|
1 |
|
|
2 |
|
|
3 |
|
||
|
1 |
Benedict Jacob |
3 |
11 |
Mathematics |
|
1 |
Mathematics |
2 |
||
|
|
|
|
12 |
Physics |
3 |
2 |
Physics |
3 |
||
|
|
|
|
13 |
Literature |
|
3 |
Literature |
4 |
||
|
|
|
|
14 |
History |
|
4 |
History |
5 |
||
|
2 |
Peter Smith |
3,5 |
21 |
Mathematics |
2 |
|
|
|
||
|
|
|
|
22 |
Physics |
|
|
|
|
||
|
|
|
|
23 |
Literature |
|
|
|
|
||
|
|
|
|
24 |
History |
5 |
|
|
|
||
|
3 |
Mary Gross |
4 |
31 |
Mathematics |
|
|
|
|
||
|
|
|
|
32 |
Physics |
|
|
|
|
||
|
|
|
|
33 |
Literature |
4 |
|
|
|
||
|
|
|
|
34 |
History |
|
|
|
|
||
|
|
|
3,5 |
|
|
3,5 |
|
|
3,5 |
||
|
Table t3 |
||||||||||
The grade book, as bookkeeping, defines the data of the previous balance sheets and its form may be the following too:
|
Knowledge-bookkeeping's database (alias grade book) | |||||||
|---|---|---|---|---|---|---|---|
|
GRADE BOOK OF CLASS 1/A |
|||||||
|
Change of knowledge-level of the student |
|
|
|
||||
|
Event's |
Event's |
Event-coordinates |
Event's |
||||
|
time-point |
Description |
Student's identifier-number |
Subject's identifier-number |
numerical value (grade) |
|||
|
data1 |
data 2 |
data 3* |
data 4 |
||||
|
|
|
y1 |
y2 |
|
|||
|
10.01.2010 |
Peter Smith answered from history |
2 |
4 |
5 |
|||
|
11.02.2010 |
Mary Gross answered from literature |
3 |
3 |
4 |
|||
|
18.03.2010 |
Peter Smith answered from mathematics |
2 |
1 |
2 |
|||
|
20.03.2010 |
Benedict Jacob wrote school exercise from physics |
1 |
2 |
3 |
|||
|
* The data3 is same with two elements y1 and y2 of the vector y=[y1,y2]* of the event-coordinates |
|||||||
|
Table t4 |
|||||||
Same time the bookkeeping of the knowledge-level happen a "grade book" of the total school too. This time clearly expedient if we record for example in the book the identity of the years (as the 5th data) and/or the identity of the classes (as the 6th data) and/or the identity of the teachers (as the 7th data). The aspect-number of the balance sheet may be this time as good 7 too; and this time we may show the conformation of the knowledge-level of the students by the teachers. This information is well usable for the managers of the schools.
Bookkeeping of the monthly telephone-cost and the monthly cost balance sheet
|
4 ASPECT | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
STATIC and DYNAMIC MONTHLY TELEPHONE COST BALANCE HEET | ||||||||||||||||
|
|
Caller number |
Phone cost $ |
|
Called number |
Phone cost $ |
|
Called cellular or wire phone |
Phone cost $ |
|
Call times |
Phone cost $ |
|||||
|
1 |
4010246 G1 |
2640 |
1 |
6308708 G |
1056 |
1 |
Cellular phone |
870 |
1 |
10.01.2010 |
650 |
|||||
|
2 |
4010247 G2 |
1056 |
2 |
3424479 H |
1570 |
2 |
Wire phone |
2951 |
2 |
11.01.2010 |
1066 |
|||||
|
3 |
4010248 G3 |
125 |
3 |
30 6649984 |
1070 |
|
|
|
3 |
12.01.2010 |
920 |
|||||
|
|
|
|
4 |
4323900 M |
125 |
|
|
|
|
13.01.2010 |
0 |
|||||
|
|
|
|
|
|
|
|
|
|
|
14.01.2010 |
0 |
|||||
|
|
|
|
|
|
|
|
|
|
4 |
15.01.2010 |
125 |
|||||
|
|
|
|
|
|
|
|
|
|
5 |
16.01.2010 |
870 |
|||||
|
|
|
|
|
|
|
|
|
|
|
17.01.2010 |
0 |
|||||
|
|
|
|
|
|
|
|
|
|
6 |
18.01.2010 |
200 |
|||||
|
|
Total sum |
3821 |
|
Total sum |
3821 |
|
Total s. |
3821 |
|
Total sum |
3831 |
|||||
|
table f1 |
||||||||||||||||
|
Monthly telephone-cost's bookkeeping database | ||||||||||||||||
|
LIST OF CALLS |
||||||||||||||||
|
Call's |
Description of the calls |
Event-coordinates |
Phone costs |
|||||||||||||
|
time |
|
Caller n. |
Called n. |
Called cell. or wire phone |
$ |
|||||||||||
|
data 1 |
data 2 |
data 3* |
data4 |
|||||||||||||
|
|
|
y1 |
y2 |
y3 |
|
|||||||||||
|
10.01.2010 |
4010246 G1 called 3424479 H |
1 |
2 |
2 |
650 |
|||||||||||
|
11.01.2010 |
4010247 G2 called 6308708 G |
2 |
1 |
2 |
1066 |
|||||||||||
|
12.01.2010 |
4010246 G1 called 3424479 H |
1 |
2 |
2 |
920 |
|||||||||||
|
15.01.2010 |
4010248 G3 called 4323900 M |
3 |
4 |
2 |
125 |
|||||||||||
|
16.01.2010 |
4010246 G1 called 30 6649984 |
1 |
3 |
1 |
870 |
|||||||||||
|
18.01.2010 |
4010246 G1 called 30 6649984 |
1 |
3 |
2 |
200 |
|||||||||||
|
Total sum:: |
3831 |
|||||||||||||||
|
* The data3 is same with three elements y1, y2 and y3 of the vector y=[y1,y2,y3]* of the event-coordinates |
||||||||||||||||
|
table f2 |
||||||||||||||||
An to day used hungarian, english, and a german classic balance sheet of property
|
KLASSZIKUS VAGYONMÉRLEG | |
|---|---|
|
(VBR) |
|
|
Eszközök (Aktívák) |
Források, vagy Tőke (Passzívák) |
|
A. Befektetett eszközök |
D. Saját forrás (vagy saját tőke) |
|
Immateriális javak (vagyoni értékű jogok) |
Jegyzett tőke vagy alapítói tőke |
|
Tárgyieszközök (telek, épület, gép, berendezés) |
Tőketartalék |
|
Tartós pézügyi befektetések |
Eredménytartalék |
|
Beruházások (épület, gép, stb.) |
Előző évek áthozott vesztesége |
|
B. Forgóeszközök |
Mérleg szerinti (tárgyidőszaki) eredmény: [(+) nyer eség v. (-) veszteség] |
|
Készletek (anyag, áru, stb.) |
E. Céltartalékok |
|
Követelése k (vevőktől, állantól, stb) |
F. Idegen forrás (vagy idegen tőke) |
|
Értékpapirok (forgatási célú) |
Hosszúlejáratú kötelezettségek |
|
Pénzeszközök (bankszámlapénz, készpénz) |
Rövidlejáratú kötelezettségek |
|
C. Aktív időbeli elhatárolások |
G. Passzív időbeli elhatárolások |
|
Eszközök összesen (A) |
Források összesen (P) |
|
E=F (A=P) |
|
|
hungarian |
|
|
CLASSICAL PROPERTY BALANCE SHEET | |
|
(PGR) |
|
|
Assets (Active) |
Capitals (Passive) |
|
A. Current assets |
A. Current liabilities |
|
Cash on hand in Banks |
Accounts payable |
|
Time deposits and short-term investments |
Notes payable |
|
Inventories |
Customer deposits |
|
Prepayments |
Taxes payable |
|
B. Fixed assets |
Interest payable |
|
Lands |
B. Long-term liabilities |
|
Buildings |
Notes payable |
|
Furniture and equipment |
C. EQUITY (other net property) |
|
Less: Accumulated depreciation |
Retained earnings |
|
C. Other assets |
Memberships |
|
Total assets (A) |
Total capitals (Equities and liabilities) (P) |
|
A=P |
|
|
english |
|
|
KLASSISCH VERMÖGEN-BILANZ | |
|---|---|
|
(VGR) |
|
|
Aktiva |
Passiva |
|
Form und Liquidität des Vermögens: |
Herkunft und Fristigkeit der finanziellen Mittel: |
|
A. Anlagevermögen (langfristig) |
A. Eigenkapital |
|
I. Immaterielle Vermögensgegenstände |
I. Gezeichnetes Kapital |
|
II. Sachanlagen |
II. Kapitalrücklage |
|
III. Finanzanlagen |
III. Gewinnrücklagen |
|
B. Umlaufvermögen (kurzfristig) |
IV. Gewinn-/Verlustvortrag |
|
I. Vorräte (Roh-, Hilfs-, und Betriebsstoffe, usw.) |
V. Jahresüberschuss/Jahresfehlbetrag |
|
II. Forderungen und sonstige Vermögensgegenstände |
B. Rückstellungen |
|
III. Wertpapiere |
C. Verbindlichkeiten (Fremdkapital) |
|
IV. Flüssige Mittel (Schecks, Kassenbestand, Bank) |
-(Langfristige Verbindlichkeiten (Schulden)) |
|
|
-(Kurzfristige Verbindlichkeiten (Schulden)) |
|
C. Rechnungsabgrenzungsposten |
D. Rechnungsabgrenzungsposten |
|
Gesamtvermögen (A) |
Gesamtkapital (P) |
|
A=P |
|
|
german |
|
The facts of experiment of a publication
1. E-mail
(2009. augusztus 31, hétfő, 4:29 PM)From:
Vida Krisztina [mailto:Vida.Krisztina@akkrt.hu]
Tisztelt Gulyás István!
Hivatkozással imént egyeztetésünkre küldök Önnek egy levelet.
Kérem, hogy kiadásra felajánlani kívánt művét válaszában szíveskedjék eljuttatni.
Köszönettel és üdvözlettel:
Vida Krisztina
Akadémiai Kiadó
Tel.: 1/464-8254
Fax: 1/464-8251
2. E-mail (2009. augusztus 31, hétfő, 5:00 PM):
From:
gulyas@ginprofessional.hu [mailto:gulyas@ginprofessional.hu]
Tisztelt Vida Krisztina!
Mellékeltem ajánlott, most elkészült könyvem online változatát PDF formátumban. Mint említettem a könyv a Modern Könyvviteltan (2009) főcímet viseli. Közelebbről:
Gulyás István
MODERN KÖNYVVITELTAN
A modern n-szeres (n
ł3) vagyonkönyvvitel,mint
az egyik speciális könyvvitel
elméletének elemei
és
axiomatikus rendszere
(a számlaelméletek halála)
2009
A tartalomjegyzék és az előszó önmagáért beszél. Külön ismertető még itt olv
asható: http://www.ginprofessional.hu/GI-A_modern_n-szeres_kvitel_20090814_01Rv_html-ben/N-szeres-kvitel_ajanl_2009_hu.html .A könyvnek van már ISBN száma, a nevemen és a cégem nevén (ez csak a jogbiztonság okán van és nem jelenti azt, hogy magam óhajtom kiadni), továbbá letétbe van helyezve az Artisjus irodánál.
Szerintem ilyen tartalommal a világon nem volt és nincs könyvviteltan. Ez a könyvviteltan új, mint például Einstein relativitáselmélete, vagy mint Bolyai abszolút geometriája az Appendixben. És Amint 2300 éve megalapozta Euklidész Elemek című műve az axiomatikus és deduktív geometriát, akként alapozza meg e könyv a könyvviteli elemek résszel a könyvviteltant és teszi több, mint 2000 év után - a matematikához hasonlóan - egzakt és deduktív - valódi - tudománnyá, a hagyományos könyvviteltankönyvek mintapéldáiból álló "receptgyűjteménye helyett". Emellett kontírozási algoritmussal segíti a gyakorló könyvelőket, és az absztrakt könyvelőautomata résszel pedig a könyvviteli szoftverek íróit. Tehát munkaeszköz is egyben! A 435 oldalból 300 népszerűsítő jelleggel íródott, a 300-435 oldal közötti rész a tudomány iránti igényt elégíti ki. Itt olvasható a könyvviteli elemek axiomatikus rendszere.
Várom mielőbbi válaszukat: érdekli-e Önöket e könyv bel-
és külföldi kiadása.
Üdvözlettel:
Gulyás István
közgazdász
3. E-mail (2009. szeptember 03, csütörtök, 1:11 PM):
From:
Fehér Katalin [mailto:Feher.Katalin@akkrt.hu]
Tisztelt Gulyás István,
köszönettel megkaptuk kéziratának elektronikus változatát.
Kiadónk a témában tanácsadói rendszerrel dönt arról, hogy mely művek gondozása illes
zkedik a kiadói profilba.Kérem szíves hozzájárulását ahhoz, hogy tanácsadóinknak továbbíthassuk az Ön által küldött levelet és kéziratot a döntés meghozatalához..
Köszönettel és üdvözlettel,
Dr. Fehér Katalin PhD
vezető szerkesztő
Akadémiai Kiadó Zrt.
1117 Budapest,
Prielle Kornélia u. 19/D
(1) 464 8274
4. E-mail
(2009. szeptember 03, csütörtök, 1:38 PM):From:
gulyas@ginprofessional.hu [mailto:gulyas@ginprofessional.hu]
Tisztelt Dr. Fehér Katalin!
Köszönöm a visszaigazolást.
A hozzájárulásomat természetesen megadom - e procedúrával ugyanis eleve számoltam.
Üdvözlettel:
Gulyás István
5. E-mail (2009. szeptember 08, kedd, 10:49 AM):
From:
GIN Professional Kft [mailto:ginprofessional@mail.datanet.hu] On Behalf Of gulyas@ginprofessional.hu
Dr. Fehér Katalin PhD
vezető szerkesztő
Akadémiai Kiadó Zrt.
Tisztelt Fehér Katalin!
A múlt hét csütörtökön folytatott telefonbeszélgetésünk alkalmával említette, hogy a ki
adó, a könyvem esetleges gondozását illető döntéséhez, közgazdász tanácsadók véleményére fog támaszkodni.Könyvem, bár tárgya szerint valóban a közgazdaságtan egyik ágával, a számvitel ré
szét képező könyvviteltannal foglalkozik, azonban a tárgyalt vagyonkönyvvitelt - először a történelem folyamán - egzakt, deduktív (azaz: alap és definiált fogalmak, valamint axiómák alkalmazásával tételeket megfogalmazó és azokat bizonyító) tudománnyá teszi a "könyvviteli elemek" axiomatikus rendszerével (320-420. oldal). Következésképpen, ez utóbbi tényt is tekintve, nem csak közgazdaságtudományi, hanem matematikai, ill. matematikai logikai tudományos mű is egyben.Tekintettel arra, hogy ez idáig - a történelem folyamán - a könyvviteltanban az axiomati
kus rendszereket a tudomány megalapozására sehol a világon nem alkalmazták, ezért (érthetően) a közgazdászokkal, ill. a közgazdász tudományos kutatókkal a képzésük során nem is ismertették azt meg. (Én magam is, mint közgazdász, autodidakta képeztem magam e témakörben.) Az axiomatikus rendszereket eleddig főképp (az arisztotelészi logikát és az elméleti fizika einsteini relativitáselméletét tekinthetjük ismert kivételnek) a matematika különféle ágainak megalapozására használták, úgymint a geometria, a halmazelmélet, a valószínűség számítás, a természetes számok peanoi elmélete, stb. Mindezek alapján úgy gondolom, hogy könyvem kiadói megítéléséhez az axiomatikus rendszerek terén kellő szakismerettel és jártassággal nem rendelkező közgazdász szakértők egymagukban nem valószínű, hogy autentikus véleményt tudnak mondani. Ezért - a Kiadó üzleti és a magam írói érdekét is szem előtt tartva - tisztelettel javasolom, hogy a könyv Akadémia Kiadói gondozásra alkalmas voltának megítéléséhez, a közgazdász tanácsadók mellett, az axiomatikus rendszerek terén kellő szakismerettel és jártassággal rendelkező (matematikus és/vagy logikai) szaktanácsadót is, ha tehetik, vegyenek igénybe.Tisztelettel és üdvözlettel:
Gulyás István
közgazdász - matematikus
6. E-mail (2009. szeptember 08, kedd, 11:26 AM):
From:
Fehér KatalinTo:
'gulyas@ginprofessional.hu'Sent: Tuesday, September 08, 2009 11:26 AM
Subject: RE: Javaslat matematikus tanácsadó bevonására
Tisztelt Gulyás István, köszönöm a hasznos információkat, üdvözlettel, Katalin
Dr. Fehér Katalin PhD
vezető szerkesztő
Akadémiai Kiadó Zrt.
1117 Budapest,
Prielle Kornélia u. 19/D
(1) 464 8274
7. E-mail
(2009. szeptember 08, kedd, 3:58 PM):From:
Fehér KatalinTo:
'gulyas@ginprofessional.hu'Sent: Tuesday, September 08, 2009 3:58 PM
Subject: RE: MODERN KÖNYVVITELTAN - ajánlat kiadásra
Tisztelt Gulyás István, ezúton értesítem, hogy
szakértői döntés alapján az Akadémiai Kiadó nem kíván élni a kézirat gondozásának jogával.Megértését köszönöm, a jövőbeli terveihez a legjobbakat kívánom. Üdvözlettel,
Dr. Fehér Katalin PhD
veze
@ Approximately equal
³ Greater or equal
£ Less or equal
Less or greater or equal
D Small difference
¥ Infinite (large or small) number or quantity
¦ Function, mapping or rule of the assignment
j Function, mapping or rule of the assignment
Implication
± Positive or negative
¹ Non equal
º Equivalent (other: same)
» Approximately same
... Continuation by previous
È Union of sets
Ë Non part of something
Ì Real part of something
Í Part or equal of something
Î Element of something
Ï Non element of something
å Sum
A The set A
S Matrix
v* Row vector
v Column vector
1 summarizing column vector (all element of it is 1)
0 null vector (all element of it is 0)
Ù Conjunction (logical and)
Ú Disjunction (logical and/or)
Ñ exclusive or (logical only or)
NOTES
1. See the total book (438 p.) in Hungarian language in the Hungarian National Library (HNL=OSZK) (
http://www.oszk.hu/index_hu.htm) or in the Central Library of Corvinus University of Budapest (www.lib.uni-corvinus.hu) or in the Library of University of Pécs (http://www.lib.pte.hu/) or in the National Library of University of Debrecen (http://www.lib.unideb.hu/) or in the Library of Hungarian Central Statistics Office (HCSO) (http://konyvtar.ksh.hu/index.htm); additionally this third part of the book is free downloadable online from the home page of the Hungarian Electronic Library (HEL=MEK) here: (http://mek.oszk.hu/07300/07350/). [BACK] http://www.ginprofessional.hu/GI-A_modern_n-szeres_kvitel_20090814_01Rv_html-ben/N-szeres-kvitel_I_ajanl_2009_hu.html) and the 3rd part may free download. [BACK]3. Gábor Szász: The axiomatic method (Textbook Publisher, Budapest, 1972), on the 20th page. [BACK]
4. Luca Paccioli: "Everything about Arithmetic, Geometry, and Proportions." on 24th page in the 11th tractate of the 3rd main part (Venice, 1494). [BACK]
5. Luca Paccioli: "Everything about Arithmetic, Geometry, and Proportions." on 24th page in the 11th tractate of the 3rd main part (Venice, 1494). [BACK]
6. Schä r, Johann, Friedrich: "Buchaltung und Bilanz", Berlin, 1914, 1919. [BACK]
7. Schmalanbach, Eugen: Dynamische Bilanz, 1933, Leipzig; Kosiol, Erich: Pagatorische Bilanz, 1976, Berlin. [BACK]
8. Disjoint (sets/subsets) = these are such sets/subsets what have not common element. [BACK]
C may be a set of existent or nonexistent (e.g.: fictitious) things. E.g.: If C is the range of some genus idea, then the output {C1,C2,...,Cn}of CRe is the set of subgenus ideas (aka: species ideas) and these subgenus ideas create the range of genus idea. This time the function CRe corresponds to the conventional logic division, respectively at iteration to classification, where Re is basic of the division or rather the classification. If now C is the set of some school's students, then the output {C1, C2,...,Cn}of CRe is the set of the school's classes, where the subset Ci is the set of students of the class i (i=1,2,...,n), if Re dictates this. Nota bene! If input C is set of existent things, then the output classes are too. Else if input C is set of nonexistent things, then output classes are too, other cases, evidently, are not possible. [BACK]10. Aspect is in viewpoint sense, in this theory. [BACK]
11. If an interval is closed from right, then it is denoted e.g. (0;t], where the left bracket is round, while the right bracket is square, and. t>0. Additionally: t is end plus part of interval, but 0 is not. [BACK]
12. Remember: this word 'or' is exclusive or. See the footnote 3. [BACK]
13. See for example in the appendix 1 the table y3 and its diagrams. [BACK]
14. Attribute is that feature of some thing or set of things or rather some occurrence which is inseparable from it and without which it may not exist and it is unthinkable too. [BACK]
15. The function of the impossible event is like as the function of the condition x¹ 0 near the 1/x. For example impossible events are the followings: from nothing take away something, or: from a negative quantity take away an in absolute value greater negative quantity so that the result is not positive. [BACK]
16. The event coordinate n-tuple shows the coherence of the ledger accounts apropos of some event in the traditional bookkeeping; that is it shows what accounts are changing. [BACK]
17. The gross property also known as total property; it is in essence itself the any not reduced property. It has more valid definitions which in differing way approximate the essence. Example, in view of definition of the net property, one definition may be the next: The gross property is in same time point and by same measure expressed algebraic sum of the equity and the liability. (See also in theorem 5.) [BACK]
18. I use the word 'or' in this work always 'exclusive or' in sense opposite the expression 'and/or', whose local force I sign always plus. [BACK]
19. In this doctrine the property is undefined base idea. We may say on the property that it is the totality of own goods of the economist what have monetary values and they are marketable (other sellable) goods. These may be material and immaterial goods (we called the latter with other word: to rights). [BACK]
20. In this doctrine, the object of property is undefined base idea such as e.g. the next ideas: property, debt/liability, claim, debtor, lender, set/totality, subset, element of set, disjoint, union, intersection, empty set, equivalence relation, function, value of function, measure, quantity, monetary value, difference/balance, economic event, and economist. [BACK]
21. Under the economist (this is undefined base idea) we understand - to sum up - all economic forms, any establishment together with: natural and artificial persons, household of a family, furthermore: farmers, managers, and all private and state enterprises, companies, etc., in other words: everybody who has property and/or liability (debt). [BACK]
22. Nota bene! In this doctrine, that let it be exact theory system, under the liabilities (this is undefined base idea) we understand totality of whatever debits, bat not include the equities idea like it is included sometimes in the traditional bookkeeping in the UK/US. [BACK]
23. The liability or other the debt is undefined base idea in this doctrine. It is onerous title of some economist which bears a relation to e.g. money or other object pass or rather else consideration completion. Nota bene! This onerous title is not marketable (or other: it is not sellable) because anybody buys claim but only crazy man buys debt J . [BACK]
24. Net property is also known as equity or eigen capital (in US/UK), or rather eigen capital or own property (in central and east EU) in the traditional bookkeeping. [BACK]
25. I summarize in this book with the "economy" word the meanings of such various words as for example the 'farming', the "agriculture", the "housekeeping", the "husbandry", the "economization", the "
industrial-economy" and the "money-economy", etc., additionally activity of the managers and all private and state enterprises, companies, etc. We may say that the economy is yon activity of the economist when it augments or use up its property some target on access or rather if it simply leaves that on itself. This latter obviously is the possible worst variant of economy. [BACK]26. The assets are named to means of the economy in the central and east EU. [BACK]
CA be the static base class of assets. CA contains the objects of the gross property in a given time t (t is a natural number) i.e.: one pound sterling Ł, two dollars $1, $2, one euro €, one pack paper p, two buildings b1and b2, three lands l1, l2 and l3. Thus we can define the assets base class by CA ={Ł, $1, $2, €, p, b1, b2, l1, l2, l3}. (Nota bene! We may express the assets in quantity or in monetary value of objects like traditionally.) The first step of the static property classification: let RCA be a classification aspect according to assets' destiny it defined on CA. RCA is expressed by 'x Î CA is current asset'. Thus the classification CRCA with respect to RCA is expressed symbolically by {CCA, CNCA}=¦ (CA, RCA)=CRCA, where the part of result CCA={Ł, $1, $2, €, p} is the class of the current assets. Now evidently: CNCA is the reminder of CA and CNCA={b1, b2, l1, l2, l3}. The CNCA may be other denoted e.g. CFA and its name is fixed assets. But the classification is repeatable. The next step may be the classification of CCA, defined by RCM. RCM is expressed by 'y Î CCA is money'. Now the operation is {CM, CNM}=¦ (CCA, RCM)=CRCM., where the part of result is class of the moneys CM={Ł, $1, $2, €} while naturally: CNM is the reminder of CCA, that is CNM={p}. CNM is called class of the stocks, it is denoted other CS. We can continue on the classification. The next step may be the classification of CFA, defined by RCB. RCB is expressed by 'z Î CFA is building'. Now the operation is {CB, CNB}=¦ (CFA, RCB)=CRCB where the part of result is class of the buildings CB={b1, b2} while naturally: CNB=CL is the class of the lands ('L' denotes 'the lands'), it is the reminder of CFA, that is CL={l1, l2, l3}. So now the outcome of the static property classification CA contains the following two middle classes of the assets: CCA, CFA, plus four final classes of the assets: CM, CS, CB, CL. To these are true: CMÇ CS=Æ , CBÇ CL=Æ or rather CCA=CMÈ CS and CFA=CBÈ CL and CCAÇ CFA=Æ and CA=CCAÈ CFA. Finally the fallow statement is also true: CMÈ CSÈ CSÈ CL=CA. [BACK]28. The capitals are named as the sources of the property in the central and east EU. [BACK]
29. Mind you! In this doctrine, that let it exact, under the liabilities (this is undefined base idea) we understand totality of whatever debits, bat not include the equities idea like it is included in the traditional bookkeeping in the UK/US. [BACK]
CC be the static base class of capitals. But CC now also contains objects of the gross property in a given time t (t is a natural number) as before, i.e.: one pound sterling Ł, two dollars $1, $2, one euro €, one pack paper p, two buildings b1and b2, or rather three lands l1, l2 and l3. This time we can define the base class of capitals by CC={Ł, $1, $2, €, p, b1, b2, l1, l2, l3}, where let the Ł, $1, € in class CC be foreign capital or rather let the one euro € be is long-term liability, but other debit is not. Let the other dollar $2 in class CC be share capital like money capital, while p and b1 are apported things. (Nota bene! We may express the capitals in quantity or in monetary value of objects like traditionally.) In first step let RFC be a classification aspect according to property objects' source, which defined on CC. This time RFC is expressed by 'x Î CC is foreign capital alias liability'. Thus the static property classification CRFC with respect to RFC is expressed symbolically by {CFC, CNFC}=¦ (CC, RFC)=CRFC., where the part of result CFC={Ł, $1, €} is the class of the foreign capitals alias liabilities. Evidently that CNFC is the reminder of CC and CNFC={$2, p, b1, b2, l1, l2, l3}. The class CNFC may be other denoted e.g. CEC and its name is eigen capitals alias equity. This classification is also repeatable. The second step may be the classification of CFC, defined by RFCL. RFCL is expressed by 'y Î CFC is long-term liability'. Now the operation is {CFCL, CNFCL}=¦ (CFC, RFCL)=CRFCL., where the part of result is class of the long-term liabilities CFCL={€} while naturally: CNFCL is the reminder of CFC, that is CNFCL={Ł, $1}, where CNFCL is called class of the current liability, it is denoted CFCC. We can continue on the classification. The third step may be the classification of CEC, defined by RECS. RECS is expressed by 'z Î CEC is share capital'. Now the operation is {CECS, CNECS}=¦ (CEC, RECS)=CRECS where the part of result is class of the share capital CECS={$2, p, b1} while naturally: CNECS=CECO is the class of the other eigen capitals ('O' denotes 'other'), it is the reminder of CEC, that is CECO={b2, l1, l2, l3}. The fourth step may be the classification of CECS, defined by RECSA. RECSA is expressed by 'u Î CECS is apport part of share capital'. The operation is {CECSM, CNECSM}=¦ (CECS, RECSA)=CRECSA where the part of result is class of the money of share capital CECSM={$2} while CNECS=CECSA is the class of the apport of share capital ('A' denotes 'apport'), it is the reminder of CECS, that is CECSA={p, b1}. So now the outcome of the static property classification CA contains the following three middle classes of the capitals: CFC, CEC and CECS plus five final classes of the capitals: CFCL, CFCC, CECSM, CECSA and CECO. The fallow statements are true: the pairs of the final classes are empty together with: CC=CFCÈ CEC, CFC=CFCLÈ CFCC, CEC=CECSÈ CECO and CECS=CECSMÈ CECSA. Finally: the statement is also true: CFCLÈ CFCCÈ CECSMÈ CECSAÈ CECO=CC. [BACK] CCH,(0,1] be class of the property change in the interval (0;1], that is CCH,(0,1]={Ł, $1, $2, $3, €, p, b1}, cf. example 1 and 2. Let CCH,(1,2] be class of the property change in the interval (1,2], that is CCH,(1,2]={-$3, b2, l1, l2, l3}. Here '-$3' is a property object which emerged from the property and '-' symbolizes case of the emergence. Furthermore let CCH,(0,2] be class of the property change in the interval (0;2], but CCH,(0,2]=CCH,(0,1]È CCH,(1,2], cf. definition 19, and thus we may write CCH,(0,2]={Ł, $1, $2, $3, -$3, €, p, b1, b2, l1, l2, l3}. Now let CDECR,(0,2] be class of the property decrease in the interval (0;2], that is: CDECR,(0,2]={$3, -$3}. Consequently the fallow express is true by definition 9, 10 and 11: CCH,(0,2] - C DECR,(0,2]=CBAL,(0,2]={Ł, $1, $2, €, p, b1, b2, l1, l2, l3}. However the fallow express also is true: CBAL,(0,2]=CA,2={Ł, $1, $2, €, p, b1, b2, l1, l2, l3}, where CA,2 is a static property class in time 2 of the interval (0;2] and it is equal to the balance class CBAL,(0,2], cf. def. 11. [BACK]32. All costs are expenses, sooner or later, but not all expenses (such as those incurred in acquisition of an income-generating asset) are costs. [BACK]
33. The loss is undefined base idea in the traditional bookkeeping and it is same with the decrement of own property of the economist. The loss may materialize in whatsoever form of the property. [BACK]
34. The profit is undefined base idea in the traditional bookkeeping and it is same with the increment of own property of the economist. The profit also may materialize in whatsoever form of the property. [BACK]
35. See for example the appendix 4. [BACK]
36. The axiom is such statement of a theory system what we accept without proof to true and use the theorems of the theory on proof. Nota bene! The axiom-system of a theory may be different. There is example on this one in the geometry too. The geometric axioms of Elements of Euclid and the geometric system of Hilbert are so. This theory of the modern N-entry bookkeeping also can build up other axioms. But this time it is possible that some axiom will be verifiable theorem. I chose, after elaborate think, those axioms what are readable in this book. [BACK]
37. If this quantity of the property is monetary quantity, then clear that its notation in addition monetary value of the property too. [BACK]
38. A "P.:" utáni felsorolás azt mutatja meg, hogy az állítás mely következő tételben van premisszaként felhasználva. [BACK]
39. Naturally there is such case when there is not demand of some object of the property and hence its market-price is zero. But it has positive value as to waste evidently thus too. [BACK]
40. This axiom could be theorem also because it may proof by definitions of the dynamic and static classes just as its main and part sum. But I cut this for didactical causes. [BACK]
41. Here and later under the numbers, if it is not index, we understand always rational numbers. [BACK]
42. In the future, if it is not dubious, under the property I understand always the gross property, for the sake of the briefness. [BACK]
43. Let us agree in that: if a relation's on some side two or more variables perform, are separating with comma, then on the other side being value or expression applies to all variables one by one. [BACK]
44. The enumeration after the 'P' shows that the given axiom and/or theorem in which next theorem(s) is (are) used. [BACK]
45. The enumeration after the 'C' shows that the given theorem, as conclusion, what and how many on premise bottoms. (That is: 'C' denotes the expression 'the theorem follows from this and from this'.) [BACK]
46. Here and later under the numbers, if it is not index, we understand always rational numbers. [BACK]
47. This theorem 2 is very important, because yet in near the past too, some theoretician of the traditional bookkeeping had that: the debit (aka: liability) is negative property (assets), what is a dangerous false statement, mainly in the education. [BACK]
48. The meaning of the lemma is 'auxiliary theorem'. [BACK]
49. All costs are expenses, sooner or later, but not all expenses (such as those incurred in acquisition of an income-generating asset) are costs. [BACK]
50. Naturally, (C) is verifiable the lemma without too, in essence with mathematical induction. I(t)=I(1) may not be negative, because if I(1)<0 then it means that we took something out of nothing, what is nonsense. I(2) already may be negative, but its absolute value clear may not be greater then value of I(1). So (C) in the case t=2 is true firstly. Now we assume that the statement is true in the case t=K-1 and we verify on t=K (2≤K≤M). This time: PST(K)=PCH(K)=
I(t)+I(K)≥0 by the premise, while I(K)<0 also by the premise. But I(t)≥-I(K). Additionally, since I(K)<0 hence |I(K)|=-I(K) and so I(t)≥|I(K)| holds. That is it is true that I(K)<0 may be if 2≤K≤M, provided that I(t)≥|I(K)|. Q.e.d. However, the mathematical induction with use, we exit from the axiomatic system, because this time we do not lean on its axioms and proved theorems. Therefore we cannot choose this method here. [BACK]
51. If we leave from this theorem just as from its proof and its all premises those words what refer on the property then this theorem is same the general bookkeeping fundamental n-aspect (n³ 2) structural law too. [BACK]
52. If we leave from this theorem just as from its all premises those words what refer on the property then this theorem is same the general bookkeeping fundamental n-aspect (n³ 3) dynamic and static structural law too. [BACK]
53. Let A-C-aspect denote hereafter briefly the following expression: 'assets-capitals-aspect'. While for example the expression TA-TC-aspect equal with the next expression 'time and assets-time and capitals-aspect'. [BACK]
54. If we leave from this theorem just as from its proof and its all premises those words what refer on the property then this theorem is same the general bookkeeping fundamental n=2 attribute-aspect dynamic structural law too. [BACK]
55. If we leave from this theorem just as from its all premises those words what refer on the property then this theorem is same the general bookkeeping fundamental n=2 arbitrary attribute-aspect dynamic structural law too. [BACK]
56. Let A-C-aspect denote hereafter briefly the following expression: 'assets-capitals-aspect'. While for example the string T-TA-TC-aspect denotes the next expression 'time-time and assets-time and capitals-aspect'. [BACK]
57. The 'T-A-C-aspect' as abbreviation is same with the expression 'time-assets-capitals-aspect'. [BACK]
58. The underlined part sums are those part sums of the final classes which ones apropos of some economic event occurring change by the character of the event. I denote hereafter also with underline what is suggested for the respect. [BACK]
59. Now the event coordinates, for simplicity, denote only place of the change, here exceptionally included the coordinate tth of time class too, but their character they do not denote. [BACK]
60. The such event coordinate n-tuple is rational (other realistic) which some economist-specific economic events apropos of resulting in the economist's property classification system signs them and only them the property classes, completely, whose the part sum must change by the economic event's character and content. [BACK]
61. See the previous foot-note. [BACK]
62. The such event coordinate n-tuple is rational (other realistic) which some economist-specific economic events apropos of resulting in the economist's property classification system signs them and only them the property classes, completely, whose the part sum must change by the economic event's character and content. [BACK]
63. The 'T-TA-TC-aspect' as abbreviation is same with the expression 'time- time and assets-time and capitals-aspect'. [BACK]
64. The 'T-A-C-aspect' as abbreviation is same with the expression 'time-assets-capitals-aspect'. [BACK]
65. Here the meaning of the upper index M is 'Money'. [BACK]
66. Here the meaning of the upper index N is 'Non-money'. [BACK]
67. I understand this juristic and economic in sense. [BACK]
68. The concept, function and the production of the bookkeeping derivative are readable with examples and thorough exposition in first part of my book on the 255th-267th pages. [BACK]
69. István Peák: Introduction in the theory of the automatons I. 8th page.(Textbook Publisher, Budapest, 1977.). [BACK]
70. This vector has no geometrical meaning. [BACK]
71. The data of the event-coordinates in the traditional double bookkeeping is known as the data-pair of an account coherence which shows the alphabetical and/or numerical signs of the debit account and the credit account in the 'T ' system. [BACK]
72. You may read more information on the Mealy's abstract automatons yet in the first part of my book, in its 3.2 point of the chapter IV also. [BACK]
73. The concepts of the general bookkeeping and the special bookkeepings differentiate from each other on this point. We may class the special bookkeepings on divers methods. The either aspect of the classification may be that the bookkeeping happens by document or does not. The other aspect of the classification may be that the quantity or monetary value, etc. of elements of the base set may be or may be not negative that is: between the changes as elements may be or may be not decrement. The modern N-entry property bookkeeping, on that we discuss in this book, can rate on the class of the documented bookkeepings where in this class is decrement too (see for example the appendix 1). But for example the following special bookkeepings also rate in this class: the registration of the phone numbers (its document is the contract of the service), the bookkeeping of the library lending (its document is the ticket of the lending). The following special bookkeepings, in what there is not base document and decrement, rate on the other class. These are for example: the bookkeeping of knowledge-level in the school (data base of this one is the grade book and its balance sheet is a knowledge-level balance sheet; see for example the appendix 2); and so the bookkeeping of costs of the telephone-callings. This one results a 4-aspect that is: time-caller-called-cellular or wired phone-aspect balance sheet (see e.g. the appendix 3). But we can rate also into this class e.g. the bookkeeping of changes of a galaxy in the astronomy and the bookkeeping of the distribution of the red and yellow pages in the footballJ . [BACK]
74. These perform here as the explicit named rules of the probability theory. [BACK]
75. That is clear that here the concord of the bookkeeping's data with the reality is not verified by the documents else it would be unnecessary that we verify those by the inventory. [BACK]
76. Let us keep clear of blasphemy's show too! Hence I draw attention to following facts: (1) A bookkeeping derivative comparable with the mathematical derivative because in both cases there is all information in f(x) which relates on f'(x) and both come from f(x) with logical deduction instead of count. (2) Not only mathematical derivative exists. There is for example grammatical derivative (derivative word) and electrotechnical derivative (derivative circuit /branch-circuit) and ballistic derivative (ballistic path-deviation) too. [BACK]
77. The VAT's sum expediently may be only suggested because the transaction may be VAT-free too. This time the VAT's sum must be zero. [BACK]
78. This theory is a branching of the property theory. [BACK]
79. Under on the market seller and buyer economist I do not understand the applied or the commissioned of the economist if they act instead of economist. But if the applied or the commissioned sells its work-force or rather the service to the employer/mandatory then they and its employer/mandatory also sell the own belongings. [BACK]
80. The middleman also its own service sells away in return for poundage. The commission-agent also sells the goods as own belongings because after the selling buys that from the original owner. [BACK]