:: FINSEQ_4 semantic presentation
:: deftheorem Def1 defines is_one-to-one_at FINSEQ_4:def 1 :
theorem Th1: :: FINSEQ_4:1
canceled;
theorem Th2: :: FINSEQ_4:2
theorem Th3: :: FINSEQ_4:3
theorem Th4: :: FINSEQ_4:4
theorem Th5: :: FINSEQ_4:5
:: deftheorem Def2 defines just_once_values FINSEQ_4:def 2 :
theorem Th6: :: FINSEQ_4:6
canceled;
theorem Th7: :: FINSEQ_4:7
theorem Th8: :: FINSEQ_4:8
theorem Th9: :: FINSEQ_4:9
theorem Th10: :: FINSEQ_4:10
theorem Th11: :: FINSEQ_4:11
:: deftheorem Def3 defines <- FINSEQ_4:def 3 :
theorem Th12: :: FINSEQ_4:12
canceled;
theorem Th13: :: FINSEQ_4:13
canceled;
theorem Th14: :: FINSEQ_4:14
canceled;
theorem Th15: :: FINSEQ_4:15
canceled;
theorem Th16: :: FINSEQ_4:16
theorem Th17: :: FINSEQ_4:17
theorem Th18: :: FINSEQ_4:18
theorem Th19: :: FINSEQ_4:19
canceled;
theorem Th20: :: FINSEQ_4:20
theorem Th21: :: FINSEQ_4:21
:: deftheorem Def4 defines /. FINSEQ_4:def 4 :
for b
1, b
2 being
set for b
3 being
PartFunc of b
1,b
2for b
4 being
set holds
( b
4 in dom b
3 implies b
3 /. b
4 = b
3 . b
4 );
theorem Th22: :: FINSEQ_4:22
theorem Th23: :: FINSEQ_4:23
canceled;
theorem Th24: :: FINSEQ_4:24
theorem Th25: :: FINSEQ_4:25
theorem Th26: :: FINSEQ_4:26
theorem Th27: :: FINSEQ_4:27
:: deftheorem Def5 defines .. FINSEQ_4:def 5 :
theorem Th28: :: FINSEQ_4:28
canceled;
theorem Th29: :: FINSEQ_4:29
theorem Th30: :: FINSEQ_4:30
theorem Th31: :: FINSEQ_4:31
theorem Th32: :: FINSEQ_4:32
theorem Th33: :: FINSEQ_4:33
theorem Th34: :: FINSEQ_4:34
theorem Th35: :: FINSEQ_4:35
theorem Th36: :: FINSEQ_4:36
theorem Th37: :: FINSEQ_4:37
theorem Th38: :: FINSEQ_4:38
theorem Th39: :: FINSEQ_4:39
theorem Th40: :: FINSEQ_4:40
theorem Th41: :: FINSEQ_4:41
theorem Th42: :: FINSEQ_4:42
:: deftheorem Def6 defines -| FINSEQ_4:def 6 :
theorem Th43: :: FINSEQ_4:43
canceled;
theorem Th44: :: FINSEQ_4:44
canceled;
theorem Th45: :: FINSEQ_4:45
theorem Th46: :: FINSEQ_4:46
theorem Th47: :: FINSEQ_4:47
theorem Th48: :: FINSEQ_4:48
theorem Th49: :: FINSEQ_4:49
theorem Th50: :: FINSEQ_4:50
theorem Th51: :: FINSEQ_4:51
theorem Th52: :: FINSEQ_4:52
theorem Th53: :: FINSEQ_4:53
:: deftheorem Def7 defines |-- FINSEQ_4:def 7 :
theorem Th54: :: FINSEQ_4:54
canceled;
theorem Th55: :: FINSEQ_4:55
canceled;
theorem Th56: :: FINSEQ_4:56
canceled;
theorem Th57: :: FINSEQ_4:57
theorem Th58: :: FINSEQ_4:58
theorem Th59: :: FINSEQ_4:59
theorem Th60: :: FINSEQ_4:60
theorem Th61: :: FINSEQ_4:61
theorem Th62: :: FINSEQ_4:62
theorem Th63: :: FINSEQ_4:63
theorem Th64: :: FINSEQ_4:64
theorem Th65: :: FINSEQ_4:65
theorem Th66: :: FINSEQ_4:66
theorem Th67: :: FINSEQ_4:67
theorem Th68: :: FINSEQ_4:68
theorem Th69: :: FINSEQ_4:69
theorem Th70: :: FINSEQ_4:70
theorem Th71: :: FINSEQ_4:71
theorem Th72: :: FINSEQ_4:72
theorem Th73: :: FINSEQ_4:73
theorem Th74: :: FINSEQ_4:74
theorem Th75: :: FINSEQ_4:75
theorem Th76: :: FINSEQ_4:76
Lemma43:
for b1, b2 being finite set
for b3 being Function of b1,b2 holds
( card b1 = card b2 & rng b3 = b2 implies b3 is one-to-one )
theorem Th77: :: FINSEQ_4:77
theorem Th78: :: FINSEQ_4:78
theorem Th79: :: FINSEQ_4:79
theorem Th80: :: FINSEQ_4:80
for b
1, b
2 being
set for b
3 being
Function of b
2,b
1 holds
not (
Card b
1 <` Card b
2 & b
1 <> {} & ( for b
4, b
5 being
set holds
not ( b
4 in b
2 & b
5 in b
2 & b
4 <> b
5 & b
3 . b
4 = b
3 . b
5 ) ) )
theorem Th81: :: FINSEQ_4:81
for b
1, b
2 being
set for b
3 being
Function of b
1,b
2 holds
not (
Card b
1 <` Card b
2 & ( for b
4 being
set holds
not ( b
4 in b
2 & ( for b
5 being
set holds
not ( b
5 in b
1 & not b
3 . b
5 <> b
4 ) ) ) ) )
theorem Th82: :: FINSEQ_4:82
theorem Th83: :: FINSEQ_4:83
theorem Th84: :: FINSEQ_4:84
theorem Th85: :: FINSEQ_4:85
theorem Th86: :: FINSEQ_4:86
theorem Th87: :: FINSEQ_4:87
theorem Th88: :: FINSEQ_4:88
definition
let c
1, c
2, c
3, c
4 be
set ;
func <*c1,c2,c3,c4*> -> set equals :: FINSEQ_4:def 8
<*a1,a2,a3*> ^ <*a4*>;
correctness
coherence
<*c1,c2,c3*> ^ <*c4*> is set ;
;
let c
5 be
set ;
func <*c1,c2,c3,c4,c5*> -> set equals :: FINSEQ_4:def 9
<*a1,a2,a3*> ^ <*a4,a5*>;
correctness
coherence
<*c1,c2,c3*> ^ <*c4,c5*> is set ;
;
end;
:: deftheorem Def8 defines <* FINSEQ_4:def 8 :
:: deftheorem Def9 defines <* FINSEQ_4:def 9 :
for b
1, b
2, b
3, b
4, b
5 being
set holds
<*b1,b2,b3,b4,b5*> = <*b1,b2,b3*> ^ <*b4,b5*>;
registration
let c
1, c
2, c
3, c
4 be
set ;
cluster <*a1,a2,a3,a4*> -> Relation-like Function-like ;
coherence
( <*c1,c2,c3,c4*> is Function-like & <*c1,c2,c3,c4*> is Relation-like )
;
let c
5 be
set ;
cluster <*a1,a2,a3,a4,a5*> -> Relation-like Function-like ;
coherence
( <*c1,c2,c3,c4,c5*> is Function-like & <*c1,c2,c3,c4,c5*> is Relation-like )
;
end;
registration
let c
1, c
2, c
3, c
4 be
set ;
cluster <*a1,a2,a3,a4*> -> Relation-like Function-like FinSequence-like ;
coherence
<*c1,c2,c3,c4*> is FinSequence-like
;
let c
5 be
set ;
cluster <*a1,a2,a3,a4,a5*> -> Relation-like Function-like FinSequence-like ;
coherence
<*c1,c2,c3,c4,c5*> is FinSequence-like
;
end;
definition
let c
1 be non
empty set ;
let c
2, c
3, c
4, c
5, c
6 be
Element of c
1;
redefine func <* as
<*c2,c3,c4,c5,c6*> -> FinSequence of a
1;
coherence
<*c2,c3,c4,c5,c6*> is FinSequence of c1
end;
theorem Th89: :: FINSEQ_4:89
for b
1, b
2, b
3, b
4 being
set holds
(
<*b1,b2,b3,b4*> = <*b1,b2,b3*> ^ <*b4*> &
<*b1,b2,b3,b4*> = <*b1,b2*> ^ <*b3,b4*> &
<*b1,b2,b3,b4*> = <*b1*> ^ <*b2,b3,b4*> &
<*b1,b2,b3,b4*> = ((<*b1*> ^ <*b2*>) ^ <*b3*>) ^ <*b4*> )
theorem Th90: :: FINSEQ_4:90
for b
1, b
2, b
3, b
4, b
5 being
set holds
(
<*b1,b2,b3,b4,b5*> = <*b1,b2,b3*> ^ <*b4,b5*> &
<*b1,b2,b3,b4,b5*> = <*b1,b2,b3,b4*> ^ <*b5*> &
<*b1,b2,b3,b4,b5*> = (((<*b1*> ^ <*b2*>) ^ <*b3*>) ^ <*b4*>) ^ <*b5*> &
<*b1,b2,b3,b4,b5*> = <*b1,b2*> ^ <*b3,b4,b5*> &
<*b1,b2,b3,b4,b5*> = <*b1*> ^ <*b2,b3,b4,b5*> )
theorem Th91: :: FINSEQ_4:91
for b
1, b
2, b
3, b
4 being
set for b
5 being
FinSequence holds
( b
5 = <*b1,b2,b3,b4*> iff (
len b
5 = 4 & b
5 . 1
= b
1 & b
5 . 2
= b
2 & b
5 . 3
= b
3 & b
5 . 4
= b
4 ) )
theorem Th92: :: FINSEQ_4:92
theorem Th93: :: FINSEQ_4:93
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
FinSequence holds
( b
6 = <*b1,b2,b3,b4,b5*> iff (
len b
6 = 5 & b
6 . 1
= b
1 & b
6 . 2
= b
2 & b
6 . 3
= b
3 & b
6 . 4
= b
4 & b
6 . 5
= b
5 ) )
theorem Th94: :: FINSEQ_4:94
theorem Th95: :: FINSEQ_4:95
for b
1 being non
empty set for b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
<*b2,b3,b4,b5*> /. 1
= b
2 &
<*b2,b3,b4,b5*> /. 2
= b
3 &
<*b2,b3,b4,b5*> /. 3
= b
4 &
<*b2,b3,b4,b5*> /. 4
= b
5 )
theorem Th96: :: FINSEQ_4:96
for b
1 being non
empty set for b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
<*b2,b3,b4,b5,b6*> /. 1
= b
2 &
<*b2,b3,b4,b5,b6*> /. 2
= b
3 &
<*b2,b3,b4,b5,b6*> /. 3
= b
4 &
<*b2,b3,b4,b5,b6*> /. 4
= b
5 &
<*b2,b3,b4,b5,b6*> /. 5
= b
6 )