:: WELLORD1 semantic presentation
Lemma1:
for b1 being Relation holds
( b1 is reflexive iff for b2 being set holds
( b2 in field b1 implies [b2,b2] in b1 ) )
Lemma2:
for b1 being Relation holds
( b1 is transitive iff for b2, b3, b4 being set holds
( [b2,b3] in b1 & [b3,b4] in b1 implies [b2,b4] in b1 ) )
Lemma3:
for b1 being Relation holds
( b1 is antisymmetric iff for b2, b3 being set holds
( [b2,b3] in b1 & [b3,b2] in b1 implies b2 = b3 ) )
Lemma4:
for b1 being Relation holds
( b1 is connected iff for b2, b3 being set holds
not ( b2 in field b1 & b3 in field b1 & b2 <> b3 & not [b2,b3] in b1 & not [b3,b2] in b1 ) )
:: deftheorem Def1 defines -Seg WELLORD1:def 1 :
for b
1 being
Relationfor b
2, b
3 being
set holds
( b
3 = b
1 -Seg b
2 iff for b
4 being
set holds
( b
4 in b
3 iff ( b
4 <> b
2 &
[b4,b2] in b
1 ) ) );
theorem Th1: :: WELLORD1:1
canceled;
theorem Th2: :: WELLORD1:2
:: deftheorem Def2 defines well_founded WELLORD1:def 2 :
:: deftheorem Def3 defines is_well_founded_in WELLORD1:def 3 :
theorem Th3: :: WELLORD1:3
canceled;
theorem Th4: :: WELLORD1:4
canceled;
theorem Th5: :: WELLORD1:5
:: deftheorem Def4 defines well-ordering WELLORD1:def 4 :
:: deftheorem Def5 defines well_orders WELLORD1:def 5 :
theorem Th6: :: WELLORD1:6
canceled;
theorem Th7: :: WELLORD1:7
canceled;
theorem Th8: :: WELLORD1:8
theorem Th9: :: WELLORD1:9
for b
1 being
set for b
2 being
Relation holds
( b
2 well_orders b
1 implies for b
3 being
set holds
not ( b
3 c= b
1 & b
3 <> {} & ( for b
4 being
set holds
not ( b
4 in b
3 & ( for b
5 being
set holds
( b
5 in b
3 implies
[b4,b5] in b
2 ) ) ) ) ) )
theorem Th10: :: WELLORD1:10
theorem Th11: :: WELLORD1:11
theorem Th12: :: WELLORD1:12
theorem Th13: :: WELLORD1:13
:: deftheorem Def6 defines |_2 WELLORD1:def 6 :
theorem Th14: :: WELLORD1:14
canceled;
theorem Th15: :: WELLORD1:15
theorem Th16: :: WELLORD1:16
theorem Th17: :: WELLORD1:17
theorem Th18: :: WELLORD1:18
Lemma17:
for b1 being set
for b2 being Relation holds dom (b1 | b2) c= dom b2
theorem Th19: :: WELLORD1:19
theorem Th20: :: WELLORD1:20
theorem Th21: :: WELLORD1:21
theorem Th22: :: WELLORD1:22
theorem Th23: :: WELLORD1:23
theorem Th24: :: WELLORD1:24
theorem Th25: :: WELLORD1:25
theorem Th26: :: WELLORD1:26
theorem Th27: :: WELLORD1:27
theorem Th28: :: WELLORD1:28
theorem Th29: :: WELLORD1:29
theorem Th30: :: WELLORD1:30
theorem Th31: :: WELLORD1:31
theorem Th32: :: WELLORD1:32
theorem Th33: :: WELLORD1:33
theorem Th34: :: WELLORD1:34
canceled;
theorem Th35: :: WELLORD1:35
theorem Th36: :: WELLORD1:36
theorem Th37: :: WELLORD1:37
theorem Th38: :: WELLORD1:38
theorem Th39: :: WELLORD1:39
theorem Th40: :: WELLORD1:40
theorem Th41: :: WELLORD1:41
theorem Th42: :: WELLORD1:42
theorem Th43: :: WELLORD1:43
:: deftheorem Def7 defines is_isomorphism_of WELLORD1:def 7 :
theorem Th44: :: WELLORD1:44
canceled;
theorem Th45: :: WELLORD1:45
:: deftheorem Def8 defines are_isomorphic WELLORD1:def 8 :
theorem Th46: :: WELLORD1:46
canceled;
theorem Th47: :: WELLORD1:47
theorem Th48: :: WELLORD1:48
theorem Th49: :: WELLORD1:49
theorem Th50: :: WELLORD1:50
theorem Th51: :: WELLORD1:51
theorem Th52: :: WELLORD1:52
theorem Th53: :: WELLORD1:53
theorem Th54: :: WELLORD1:54
theorem Th55: :: WELLORD1:55
definition
let c
1, c
2 be
Relation;
assume E51:
( c
1 is
well-ordering & c
1,c
2 are_isomorphic )
;
func canonical_isomorphism_of c
1,c
2 -> Function means :
Def9:
:: WELLORD1:def 9
a
3 is_isomorphism_of a
1,a
2;
existence
ex b1 being Function st b1 is_isomorphism_of c1,c2
by E51, Def8;
uniqueness
for b1, b2 being Function holds
( b1 is_isomorphism_of c1,c2 & b2 is_isomorphism_of c1,c2 implies b1 = b2 )
by E51, Th55;
end;
:: deftheorem Def9 defines canonical_isomorphism_of WELLORD1:def 9 :
theorem Th56: :: WELLORD1:56
canceled;
theorem Th57: :: WELLORD1:57
theorem Th58: :: WELLORD1:58
theorem Th59: :: WELLORD1:59
theorem Th60: :: WELLORD1:60
theorem Th61: :: WELLORD1:61
theorem Th62: :: WELLORD1:62
theorem Th63: :: WELLORD1:63
theorem Th64: :: WELLORD1:64