:: Cardinal Arithmetics
:: by Grzegorz Bancerek
::
:: Copyright (c) 1990-2021 Association of Mizar Users

theorem :: CARD_2:1
for x being object
for X, Y being set st x in X & X,Y are_equipotent holds
( Y <> {} & ex x being object st x in Y )
proof end;

theorem :: CARD_2:2
for X being set holds
( bool X, bool (card X) are_equipotent & card (bool X) = card (bool (card X)) )
proof end;

deffunc H1( object ) -> object = $1 1 ; theorem :: CARD_2:3 for X, Y, Z being set st Z in Funcs (X,Y) holds ( Z,X are_equipotent & card Z = card X ) proof end; Lm1: for A, B being Ordinal for x1, x2 being object st x1 <> x2 holds ( A +^ B,[:A,{x1}:] \/ [:B,{x2}:] are_equipotent & card (A +^ B) = card ([:A,{x1}:] \/ [:B,{x2}:]) ) proof end; deffunc H2( set , set ) -> set = [:$1,:] \/ [:$2,{1}:]; Lm2: for X, Y being set holds ( [:X,Y:],[:Y,X:] are_equipotent & card [:X,Y:] = card [:Y,X:] ) proof end; definition let M, N be Cardinal; func M + N -> Cardinal equals :: CARD_2:def 1 card (M +^ N); coherence card (M +^ N) is Cardinal ; commutativity for b1, M, N being Cardinal st b1 = card (M +^ N) holds b1 = card (N +^ M) proof end; func M * N -> Cardinal equals :: CARD_2:def 2 card [:M,N:]; coherence card [:M,N:] is Cardinal ; commutativity for b1, M, N being Cardinal st b1 = card [:M,N:] holds b1 = card [:N,M:] by Lm2; func exp (M,N) -> Cardinal equals :: CARD_2:def 3 card (Funcs (N,M)); coherence card (Funcs (N,M)) is Cardinal ; end; :: deftheorem defines + CARD_2:def 1 : for M, N being Cardinal holds M + N = card (M +^ N); :: deftheorem defines * CARD_2:def 2 : for M, N being Cardinal holds M * N = card [:M,N:]; :: deftheorem defines exp CARD_2:def 3 : for M, N being Cardinal holds exp (M,N) = card (Funcs (N,M)); theorem Th4: :: CARD_2:4 for X, Y being set holds ( [:X,Y:],[:Y,X:] are_equipotent & card [:X,Y:] = card [:Y,X:] ) by Lm2; theorem Th5: :: CARD_2:5 for X, Y, Z being set holds ( [:[:X,Y:],Z:],[:X,[:Y,Z:]:] are_equipotent & card [:[:X,Y:],Z:] = card [:X,[:Y,Z:]:] ) proof end; Lm3: for X, Y being set holds [:X,Y:],[:(card X),Y:] are_equipotent proof end; theorem :: CARD_2:6 canceled; ::$CT
theorem Th6: :: CARD_2:7
for X, Y being set holds
( [:X,Y:],[:(card X),Y:] are_equipotent & [:X,Y:],[:X,(card Y):] are_equipotent & [:X,Y:],[:(card X),(card Y):] are_equipotent & card [:X,Y:] = card [:(card X),Y:] & card [:X,Y:] = card [:X,(card Y):] & card [:X,Y:] = card [:(card X),(card Y):] )
proof end;

theorem Th7: :: CARD_2:8
for X1, X2, Y1, Y2 being set st X1,Y1 are_equipotent & X2,Y2 are_equipotent holds
( [:X1,X2:],[:Y1,Y2:] are_equipotent & card [:X1,X2:] = card [:Y1,Y2:] )
proof end;

theorem :: CARD_2:9
for A, B being Ordinal
for x1, x2 being object st x1 <> x2 holds
( A +^ B,[:A,{x1}:] \/ [:B,{x2}:] are_equipotent & card (A +^ B) = card ([:A,{x1}:] \/ [:B,{x2}:]) ) by Lm1;

theorem Th9: :: CARD_2:10
for K, M being Cardinal
for x1, x2 being object st x1 <> x2 holds
( K + M,[:K,{x1}:] \/ [:M,{x2}:] are_equipotent & K + M = card ([:K,{x1}:] \/ [:M,{x2}:]) )
proof end;

theorem Th10: :: CARD_2:11
for A, B being Ordinal holds
( A *^ B,[:A,B:] are_equipotent & card (A *^ B) = card [:A,B:] )
proof end;

deffunc H3( set , set ) -> set = [:$1,:] \/ [:$2,{1}:];

deffunc H4( set , set , object , object ) -> set = [:$1,{$3}:] \/ [:$2,{$4}:];

theorem Th11: :: CARD_2:12
for x1, x2, y1, y2 being object
for X1, X2, Y1, Y2 being set st X1,Y1 are_equipotent & X2,Y2 are_equipotent & x1 <> x2 & y1 <> y2 holds
( [:X1,{x1}:] \/ [:X2,{x2}:],[:Y1,{y1}:] \/ [:Y2,{y2}:] are_equipotent & card ([:X1,{x1}:] \/ [:X2,{x2}:]) = card ([:Y1,{y1}:] \/ [:Y2,{y2}:]) )
proof end;

theorem Th12: :: CARD_2:13
for A, B being Ordinal holds card (A +^ B) = (card A) + (card B)
proof end;

theorem Th13: :: CARD_2:14
for A, B being Ordinal holds card (A *^ B) = (card A) * (card B)
proof end;

theorem :: CARD_2:15
for X, Y being set holds
( [:X,:] \/ [:Y,{1}:],[:Y,:] \/ [:X,{1}:] are_equipotent & card ([:X,:] \/ [:Y,{1}:]) = card ([:Y,:] \/ [:X,{1}:]) ) by Th11;

theorem Th15: :: CARD_2:16
for X1, X2, Y1, Y2 being set holds
( [:X1,X2:] \/ [:Y1,Y2:],[:X2,X1:] \/ [:Y2,Y1:] are_equipotent & card ([:X1,X2:] \/ [:Y1,Y2:]) = card ([:X2,X1:] \/ [:Y2,Y1:]) )
proof end;

theorem Th16: :: CARD_2:17
for x, y being object
for X, Y being set st x <> y holds
(card X) + (card Y) = card ([:X,{x}:] \/ [:Y,{y}:])
proof end;

theorem :: CARD_2:18
for M being Cardinal holds M + 0 = M
proof end;

Lm4: for x, y being object
for X, Y being set st x <> y holds
[:X,{x}:] misses [:Y,{y}:]

proof end;

theorem Th18: :: CARD_2:19
for K, M, N being Cardinal holds (K + M) + N = K + (M + N)
proof end;

theorem :: CARD_2:20
for K being Cardinal holds K * 0 = 0 ;

theorem Th20: :: CARD_2:21
for K being Cardinal holds K * 1 = K
proof end;

theorem Th21: :: CARD_2:22
for K, M, N being Cardinal holds (K * M) * N = K * (M * N)
proof end;

theorem Th22: :: CARD_2:23
for K being Cardinal holds 2 * K = K + K
proof end;

theorem Th23: :: CARD_2:24
for K, M, N being Cardinal holds K * (M + N) = (K * M) + (K * N)
proof end;

theorem :: CARD_2:25
for K being Cardinal holds exp (K,0) = 1
proof end;

theorem :: CARD_2:26
for K being Cardinal st K <> 0 holds
exp (0,K) = 0 ;

theorem :: CARD_2:27
for K being Cardinal holds
( exp (K,1) = K & exp (1,K) = 1 )
proof end;

theorem :: CARD_2:28
for K, M, N being Cardinal holds exp (K,(M + N)) = (exp (K,M)) * (exp (K,N))
proof end;

theorem :: CARD_2:29
for K, M, N being Cardinal holds exp ((K * M),N) = (exp (K,N)) * (exp (M,N))
proof end;

theorem :: CARD_2:30
for K, M, N being Cardinal holds exp (K,(M * N)) = exp ((exp (K,M)),N)
proof end;

:: The Number of Subsets of a Set
theorem :: CARD_2:31
for X being set holds exp (2,(card X)) = card (bool X)
proof end;

theorem :: CARD_2:32
for K being Cardinal holds exp (K,2) = K * K by ;

theorem :: CARD_2:33
for K, M being Cardinal holds exp ((K + M),2) = ((K * K) + ((2 * K) * M)) + (M * M)
proof end;

theorem Th33: :: CARD_2:34
for X, Y being set holds card (X \/ Y) c= (card X) + (card Y)
proof end;

theorem Th34: :: CARD_2:35
for X, Y being set st X misses Y holds
card (X \/ Y) = (card X) + (card Y)
proof end;

theorem Th35: :: CARD_2:36
for m, n being Nat holds n + m = n +^ m
proof end;

theorem Th36: :: CARD_2:37
for m, n being Nat holds n * m = n *^ m
proof end;

theorem Th37: :: CARD_2:38
for m, n being Nat holds card (n + m) = (card n) + (card m) by Th35;

theorem Th38: :: CARD_2:39
for m, n being Nat holds card (n * m) = (card n) * (card m)
proof end;

theorem Th39: :: CARD_2:40
for X, Y being finite set st X misses Y holds
card (X \/ Y) = (card X) + (card Y)
proof end;

theorem Th40: :: CARD_2:41
for x being object
for X being finite set st not x in X holds
card (X \/ {x}) = (card X) + 1
proof end;

theorem Th41: :: CARD_2:42
for X being set holds
( card X = 1 iff ex x being object st X = {x} )
proof end;

theorem Th42: :: CARD_2:43
for X, Y being finite set holds card (X \/ Y) <= (card X) + (card Y)
proof end;

theorem Th43: :: CARD_2:44
for X, Y being finite set st Y c= X holds
card (X \ Y) = (card X) - (card Y)
proof end;

theorem :: CARD_2:45
for X, Y being finite set holds card (X \/ Y) = ((card X) + (card Y)) - (card (X /\ Y))
proof end;

theorem :: CARD_2:46
for X, Y being finite set holds card [:X,Y:] = (card X) * (card Y)
proof end;

theorem :: CARD_2:47
for f being finite Function holds card (rng f) <= card (dom f)
proof end;

theorem :: CARD_2:48
for X, Y being finite set st X c< Y holds
( card X < card Y & card X in Segm (card Y) )
proof end;

theorem :: CARD_2:49
for X, Y being set st ( card X c= card Y or card X in card Y ) & Y is finite holds
X is finite
proof end;

theorem Th49: :: CARD_2:50
for x1, x2 being object holds card {x1,x2} <= 2
proof end;

theorem Th50: :: CARD_2:51
for x1, x2, x3 being object holds card {x1,x2,x3} <= 3
proof end;

theorem Th51: :: CARD_2:52
for x1, x2, x3, x4 being object holds card {x1,x2,x3,x4} <= 4
proof end;

theorem Th52: :: CARD_2:53
for x1, x2, x3, x4, x5 being object holds card {x1,x2,x3,x4,x5} <= 5
proof end;

theorem Th53: :: CARD_2:54
for x1, x2, x3, x4, x5, x6 being object holds card {x1,x2,x3,x4,x5,x6} <= 6
proof end;

theorem Th54: :: CARD_2:55
for x1, x2, x3, x4, x5, x6, x7 being object holds card {x1,x2,x3,x4,x5,x6,x7} <= 7
proof end;

theorem :: CARD_2:56
for x1, x2, x3, x4, x5, x6, x7, x8 being object holds card {x1,x2,x3,x4,x5,x6,x7,x8} <= 8
proof end;

theorem Th56: :: CARD_2:57
for x1, x2 being object st x1 <> x2 holds
card {x1,x2} = 2
proof end;

theorem Th57: :: CARD_2:58
for x1, x2, x3 being object st x1 <> x2 & x1 <> x3 & x2 <> x3 holds
card {x1,x2,x3} = 3
proof end;

theorem Th58: :: CARD_2:59
for x1, x2, x3, x4 being object st x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds
card {x1,x2,x3,x4} = 4
proof end;

:: from GROUP_3
theorem :: CARD_2:60
for X being set st card X = 2 holds
ex x, y being object st
( x <> y & X = {x,y} )
proof end;

:: from YELLOW_6, 2004.07.25
theorem :: CARD_2:61
for f being Function holds card (rng f) c= card (dom f)
proof end;

Lm5: now :: thesis: for n being Nat st ( for Z being finite set st card Z = n & Z <> {} & ( for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ) holds
union Z in Z ) holds
for Z being finite set st card Z = n + 1 & Z <> {} & ( for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ) holds
union Z in Z
let n be Nat; :: thesis: ( ( for Z being finite set st card Z = n & Z <> {} & ( for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ) holds
union Z in Z ) implies for Z being finite set st card Z = n + 1 & Z <> {} & ( for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ) holds
union b3 in b3 )

assume A1: for Z being finite set st card Z = n & Z <> {} & ( for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ) holds
union Z in Z ; :: thesis: for Z being finite set st card Z = n + 1 & Z <> {} & ( for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ) holds
union b3 in b3

let Z be finite set ; :: thesis: ( card Z = n + 1 & Z <> {} & ( for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ) implies union b2 in b2 )

assume that
A2: card Z = n + 1 and
A3: Z <> {} and
A4: for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ; :: thesis: union b2 in b2
set y = the Element of Z;
per cases ( n = 0 or n <> 0 ) ;
suppose n = 0 ; :: thesis: union b2 in b2
then consider x being object such that
A5: Z = {x} by ;
thus union Z in Z by ; :: thesis: verum
end;
suppose A6: n <> 0 ; :: thesis: union b2 in b2
set Y = Z \ { the Element of Z};
reconsider Y = Z \ { the Element of Z} as finite set ;
{ the Element of Z} c= Z by ;
then A7: card Y = (n + 1) - (card { the Element of Z}) by
.= (n + 1) - 1 by CARD_1:30
.= n ;
for a, b being set st a in Y & b in Y & not a c= b holds
b c= a by A4;
then A8: union Y in Y by ;
then A9: union Y in Z ;
Z = (Z \ { the Element of Z}) \/ { the Element of Z}
proof
thus Z c= (Z \ { the Element of Z}) \/ { the Element of Z} :: according to XBOOLE_0:def 10 :: thesis: (Z \ { the Element of Z}) \/ { the Element of Z} c= Z
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Z or x in (Z \ { the Element of Z}) \/ { the Element of Z} )
assume x in Z ; :: thesis: x in (Z \ { the Element of Z}) \/ { the Element of Z}
then ( x in Z \ { the Element of Z} or x in { the Element of Z} ) by XBOOLE_0:def 5;
hence x in (Z \ { the Element of Z}) \/ { the Element of Z} by XBOOLE_0:def 3; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (Z \ { the Element of Z}) \/ { the Element of Z} or x in Z )
assume x in (Z \ { the Element of Z}) \/ { the Element of Z} ; :: thesis: x in Z
then A10: ( x in Z \ { the Element of Z} or x in { the Element of Z} ) by XBOOLE_0:def 3;
{ the Element of Z} c= Z by ;
hence x in Z by A10; :: thesis: verum
end;
then A11: union Z = () \/ (union { the Element of Z}) by ZFMISC_1:78
.= () \/ the Element of Z ;
A12: the Element of Z in Z by A3;
( the Element of Z c= union Y or union Y c= the Element of Z ) by A4, A8;
hence union Z in Z by ; :: thesis: verum
end;
end;
end;

Lm6: for Z being finite set st Z <> {} & ( for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ) holds
union Z in Z

proof end;

theorem :: CARD_2:62
for Z being set st Z <> {} & Z is finite & ( for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ) holds
union Z in Z by Lm6;

theorem :: CARD_2:63
for x1, x2, x3, x4, x5 being object st x1,x2,x3,x4,x5 are_mutually_distinct holds
card {x1,x2,x3,x4,x5} = 5
proof end;

:: from MATRIX_1, 2007.07.22, A.T, generalized
theorem :: CARD_2:64
for M1, M2 being set st card M1 = 0 & card M2 = 0 holds
M1 = M2
proof end;

:: missing, 2007.06.14, A.T.
registration
let x, y be object ;
cluster [x,y] -> non natural ;
coherence
not [x,y] is natural
proof end;
end;

theorem :: CARD_2:65
for M, N being Cardinal holds Sum (M --> N) = M * N
proof end;

theorem :: CARD_2:66
for M, N being Cardinal holds Product (N --> M) = exp (M,N) by CARD_3:11;

scheme :: CARD_2:sch 1
FinRegularity{ F1() -> finite set , P1[ object , object ] } :
ex x being object st
( x in F1() & ( for y being object st y in F1() & y <> x holds
not P1[y,x] ) )
provided
A1: F1() <> {} and
A2: for x, y being object st P1[x,y] & P1[y,x] holds
x = y and
A3: for x, y, z being object st P1[x,y] & P1[y,z] holds
P1[x,z]
proof end;

scheme :: CARD_2:sch 2
MaxFinSetElem{ F1() -> finite set , P1[ object , object ] } :
ex x being object st
( x in F1() & ( for y being object st y in F1() holds
P1[x,y] ) )
provided
A1: F1() <> {} and
A2: for x, y being object holds
( P1[x,y] or P1[y,x] ) and
A3: for x, y, z being object st P1[x,y] & P1[y,z] holds
P1[x,z]
proof end;

Lm7: for n being Nat st Rank n is finite holds
Rank (n + 1) is finite

proof end;

Lm8: 1 = card 1
;

theorem :: CARD_2:67
for n being Nat holds Rank n is finite
proof end;

theorem :: CARD_2:68
for M being Cardinal holds
( 0 in M iff 1 c= M )
proof end;

theorem :: CARD_2:69
for M being Cardinal holds
( 1 in M iff 2 c= M )
proof end;

theorem Th69: :: CARD_2:70
for A being Ordinal holds
( A is limit_ordinal iff for B being Ordinal
for n being Nat st B in A holds
B +^ n in A )
proof end;

theorem Th70: :: CARD_2:71
for A being Ordinal
for n being Nat holds
( A +^ (succ n) = (succ A) +^ n & A +^ (n + 1) = (succ A) +^ n )
proof end;

theorem Th71: :: CARD_2:72
for A being Ordinal ex n being Nat st A *^ (succ 1) = A +^ n
proof end;

theorem Th72: :: CARD_2:73
for A being Ordinal st A is limit_ordinal holds
A *^ (succ 1) = A
proof end;

Lm9:
by ORDINAL1:def 11;

theorem Th73: :: CARD_2:74
for A being Ordinal st omega c= A holds
1 +^ A = A
proof end;

registration
coherence
for b1 being Cardinal st b1 is infinite holds
b1 is limit_ordinal
proof end;
end;

theorem Th74: :: CARD_2:75
for M being Cardinal st not M is finite holds
M + M = M
proof end;

theorem Th75: :: CARD_2:76
for M, N being Cardinal st not M is finite & ( N c= M or N in M ) holds
( M + N = M & N + M = M )
proof end;

theorem :: CARD_2:77
for X, Y being set st not X is finite & ( X,Y are_equipotent or Y,X are_equipotent ) holds
( X \/ Y,X are_equipotent & card (X \/ Y) = card X )
proof end;

theorem :: CARD_2:78
for X, Y being set st not X is finite & Y is finite holds
( X \/ Y,X are_equipotent & card (X \/ Y) = card X )
proof end;

theorem :: CARD_2:79
for X, Y being set st not X is finite & ( card Y in card X or card Y c= card X ) holds
( X \/ Y,X are_equipotent & card (X \/ Y) = card X )
proof end;

registration
let M, N be finite Cardinal;
cluster M + N -> finite ;
coherence
M + N is finite
proof end;
end;

theorem :: CARD_2:80
for M, N being finite Cardinal holds M + N is finite ;

theorem :: CARD_2:81
for M, N being Cardinal st not M is finite holds
( not M + N is finite & not N + M is finite )
proof end;

theorem :: CARD_2:82
for M, N being finite Cardinal holds M * N is finite ;

theorem Th82: :: CARD_2:83
for K, L, M, N being Cardinal st ( ( K in L & M in N ) or ( K c= L & M in N ) or ( K in L & M c= N ) or ( K c= L & M c= N ) ) holds
( K + M c= L + N & M + K c= L + N )
proof end;

theorem :: CARD_2:84
for K, M, N being Cardinal st ( M in N or M c= N ) holds
( K + M c= K + N & K + M c= N + K & M + K c= K + N & M + K c= N + K ) by Th82;

theorem Th84: :: CARD_2:85
for X, Y being set st X is countable & Y is countable holds
X \/ Y is countable
proof end;

theorem Th85: :: CARD_2:86
for M, N being Cardinal
for f being Function st card (dom f) c= M & ( for x being object st x in dom f holds
card (f . x) c= N ) holds
card () c= M * N
proof end;

theorem :: CARD_2:87
for M, N being Cardinal
for X being set st card X c= M & ( for Y being set st Y in X holds
card Y c= N ) holds
card () c= M * N
proof end;

theorem Th87: :: CARD_2:88
for f being Function st dom f is finite & ( for x being object st x in dom f holds
f . x is finite ) holds
Union f is finite
proof end;

theorem :: CARD_2:89
for n being Nat holds
( omega * (card n) c= omega & (card n) * omega c= omega )
proof end;

theorem Th89: :: CARD_2:90
for K, L, M, N being Cardinal st ( ( K in L & M in N ) or ( K c= L & M in N ) or ( K in L & M c= N ) or ( K c= L & M c= N ) ) holds
( K * M c= L * N & M * K c= L * N )
proof end;

theorem :: CARD_2:91
for K, M, N being Cardinal st ( M in N or M c= N ) holds
( K * M c= K * N & K * M c= N * K & M * K c= K * N & M * K c= N * K ) by Th89;

theorem Th91: :: CARD_2:92
for K, L, M, N being Cardinal holds
( ( not ( K in L & M in N ) & not ( K c= L & M in N ) & not ( K in L & M c= N ) & not ( K c= L & M c= N ) ) or K = 0 or exp (K,M) c= exp (L,N) )
proof end;

theorem :: CARD_2:93
for K, M, N being Cardinal holds
( ( not M in N & not M c= N ) or K = 0 or ( exp (K,M) c= exp (K,N) & exp (M,K) c= exp (N,K) ) )
proof end;

theorem Th93: :: CARD_2:94
for M, N being Cardinal holds
( M c= M + N & N c= M + N )
proof end;

theorem :: CARD_2:95
for M, N being Cardinal st N <> 0 holds
( M c= M * N & M c= N * M )
proof end;

theorem Th95: :: CARD_2:96
for K, L, M, N being Cardinal st K in L & M in N holds
( K + M in L + N & M + K in L + N )
proof end;

theorem :: CARD_2:97
for K, M, N being Cardinal st K + M in K + N holds
M in N
proof end;

theorem :: CARD_2:98
for X, Y being set st (card X) + (card Y) = card X & card Y in card X holds
card (X \ Y) = card X
proof end;

theorem :: CARD_2:99
for K, M, N being Cardinal st K * M in K * N holds
M in N
proof end;

theorem :: CARD_2:100
for X, Y being set st X is countable & Y is countable holds
X \+\ Y is countable
proof end;

registration
let A be finite set ;
let B be set ;
let f be Function of A,(Fin B);
coherence
Union f is finite
proof end;
end;

registration
let f be finite finite-yielding Function;
coherence
proof end;
end;

::from COMPL_SP, 2011.07.30, A.T.
theorem :: CARD_2:101
for F being Function st dom F is infinite & rng F is finite holds
ex x being object st
( x in rng F & F " {x} is infinite )
proof end;

:: from PRE_POLY, CARD_FIN etc., 2012.06.27, A.T.
theorem :: CARD_2:102
for X, Y being finite set st X c= Y & card X = card Y holds
X = Y
proof end;

scheme :: CARD_2:sch 3
Sch3{ F1() -> finite set , P1[ object , object ] } :
ex x being set st
( x in F1() & ( for y being set st y in F1() & y <> x holds
not P1[y,x] ) )
provided
A1: F1() <> {} and
A2: for x, y being set st P1[x,y] & P1[y,x] holds
x = y and
A3: for x, y, z being set st P1[x,y] & P1[y,z] holds
P1[x,z]
proof end;