:: AFF_1 semantic presentation
:: deftheorem Def1 defines LIN AFF_1:def 1 :
theorem Th1: :: AFF_1:1
canceled;
theorem Th2: :: AFF_1:2
canceled;
theorem Th3: :: AFF_1:3
canceled;
theorem Th4: :: AFF_1:4
canceled;
theorem Th5: :: AFF_1:5
canceled;
theorem Th6: :: AFF_1:6
canceled;
theorem Th7: :: AFF_1:7
canceled;
theorem Th8: :: AFF_1:8
canceled;
theorem Th9: :: AFF_1:9
canceled;
theorem Th10: :: AFF_1:10
theorem Th11: :: AFF_1:11
Lemma4:
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3 // b4,b5 implies b4,b5 // b2,b3 )
theorem Th12: :: AFF_1:12
Lemma6:
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3 // b4,b5 implies b3,b2 // b4,b5 )
Lemma7:
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3 // b4,b5 implies b2,b3 // b5,b4 )
theorem Th13: :: AFF_1:13
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies ( b
2,b
3 // b
5,b
4 & b
3,b
2 // b
4,b
5 & b
3,b
2 // b
5,b
4 & b
4,b
5 // b
2,b
3 & b
4,b
5 // b
3,b
2 & b
5,b
4 // b
2,b
3 & b
5,b
4 // b
3,b
2 ) )
theorem Th14: :: AFF_1:14
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2 <> b
3 & not ( not ( b
2,b
3 // b
4,b
5 & b
2,b
3 // b
6,b
7 ) & not ( b
2,b
3 // b
4,b
5 & b
6,b
7 // b
2,b
3 ) & not ( b
4,b
5 // b
2,b
3 & b
6,b
7 // b
2,b
3 ) & not ( b
4,b
5 // b
2,b
3 & b
2,b
3 // b
6,b
7 ) ) implies b
4,b
5 // b
6,b
7 )
Lemma10:
for b1 being AffinSpace
for b2, b3, b4 being Element of b1 holds
( LIN b2,b3,b4 implies ( LIN b2,b4,b3 & LIN b3,b2,b4 ) )
theorem Th15: :: AFF_1:15
for b
1 being
AffinSpacefor b
2, b
3, b
4 being
Element of b
1 holds
(
LIN b
2,b
3,b
4 implies (
LIN b
2,b
4,b
3 &
LIN b
3,b
2,b
4 &
LIN b
3,b
4,b
2 &
LIN b
4,b
2,b
3 &
LIN b
4,b
3,b
2 ) )
theorem Th16: :: AFF_1:16
theorem Th17: :: AFF_1:17
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 <> b
3 &
LIN b
2,b
3,b
4 &
LIN b
2,b
3,b
5 &
LIN b
2,b
3,b
6 implies
LIN b
4,b
5,b
6 )
theorem Th18: :: AFF_1:18
theorem Th19: :: AFF_1:19
theorem Th20: :: AFF_1:20
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 <> b
3 &
LIN b
4,b
5,b
2 &
LIN b
4,b
5,b
3 &
LIN b
2,b
3,b
6 implies
LIN b
4,b
5,b
6 )
theorem Th21: :: AFF_1:21
theorem Th22: :: AFF_1:22
theorem Th23: :: AFF_1:23
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 &
LIN b
2,b
4,b
5 & b
3,b
4 // b
3,b
5 implies b
4 = b
5 )
definition
let c
1 be
AffinSpace;
let c
2, c
3 be
Element of c
1;
func Line c
2,c
3 -> Subset of a
1 means :
Def2:
:: AFF_1:def 2
for b
1 being
Element of a
1 holds
( b
1 in a
4 iff
LIN a
2,a
3,b
1 );
existence
ex b1 being Subset of c1 st
for b2 being Element of c1 holds
( b2 in b1 iff LIN c2,c3,b2 )
uniqueness
for b1, b2 being Subset of c1 holds
( ( for b3 being Element of c1 holds
( b3 in b1 iff LIN c2,c3,b3 ) ) & ( for b3 being Element of c1 holds
( b3 in b2 iff LIN c2,c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines Line AFF_1:def 2 :
Lemma19:
for b1 being AffinSpace
for b2, b3 being Element of b1 holds Line b2,b3 c= Line b3,b2
theorem Th24: :: AFF_1:24
canceled;
theorem Th25: :: AFF_1:25
theorem Th26: :: AFF_1:26
theorem Th27: :: AFF_1:27
theorem Th28: :: AFF_1:28
:: deftheorem Def3 defines being_line AFF_1:def 3 :
Lemma25:
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b4 is_line & b2 in b4 & b3 in b4 & b2 <> b3 implies b4 = Line b2,b3 )
theorem Th29: :: AFF_1:29
canceled;
theorem Th30: :: AFF_1:30
theorem Th31: :: AFF_1:31
theorem Th32: :: AFF_1:32
theorem Th33: :: AFF_1:33
:: deftheorem Def4 defines // AFF_1:def 4 :
:: deftheorem Def5 defines // AFF_1:def 5 :
theorem Th34: :: AFF_1:34
canceled;
theorem Th35: :: AFF_1:35
canceled;
theorem Th36: :: AFF_1:36
theorem Th37: :: AFF_1:37
theorem Th38: :: AFF_1:38
theorem Th39: :: AFF_1:39
theorem Th40: :: AFF_1:40
theorem Th41: :: AFF_1:41
theorem Th42: :: AFF_1:42
canceled;
theorem Th43: :: AFF_1:43
theorem Th44: :: AFF_1:44
theorem Th45: :: AFF_1:45
theorem Th46: :: AFF_1:46
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1for b
6 being
Subset of b
1 holds
( b
2,b
3 // b
6 & b
2,b
3 // b
4,b
5 & b
2 <> b
3 implies b
4,b
5 // b
6 )
theorem Th47: :: AFF_1:47
theorem Th48: :: AFF_1:48
theorem Th49: :: AFF_1:49
theorem Th50: :: AFF_1:50
theorem Th51: :: AFF_1:51
theorem Th52: :: AFF_1:52
theorem Th53: :: AFF_1:53
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1for b
6, b
7 being
Subset of b
1 holds
( b
2 in b
6 & b
3 in b
6 & b
4 in b
7 & b
5 in b
7 & b
6 // b
7 implies b
2,b
3 // b
4,b
5 )
theorem Th54: :: AFF_1:54
theorem Th55: :: AFF_1:55
theorem Th56: :: AFF_1:56
theorem Th57: :: AFF_1:57
Lemma49:
for b1 being AffinSpace
for b2, b3, b4 being Subset of b1 holds
( b2 // b3 & b3 // b4 implies b2 // b4 )
theorem Th58: :: AFF_1:58
for b
1 being
AffinSpacefor b
2, b
3, b
4 being
Subset of b
1 holds
( not ( not ( b
2 // b
3 & b
3 // b
4 ) & not ( b
2 // b
3 & b
4 // b
3 ) & not ( b
3 // b
2 & b
3 // b
4 ) & not ( b
3 // b
2 & b
4 // b
3 ) ) implies b
2 // b
4 )
by Lemma49;
theorem Th59: :: AFF_1:59
theorem Th60: :: AFF_1:60
theorem Th61: :: AFF_1:61
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1for b
7 being
Subset of b
1 holds
( b
2,b
3 // b
7 & b
4,b
5 // b
7 &
LIN b
6,b
2,b
4 &
LIN b
6,b
3,b
5 & b
6 in b
7 & not b
2 in b
7 & b
2 = b
3 implies b
4 = b
5 )
theorem Th62: :: AFF_1:62
theorem Th63: :: AFF_1:63
theorem Th64: :: AFF_1:64
theorem Th65: :: AFF_1:65
theorem Th66: :: AFF_1:66
theorem Th67: :: AFF_1:67
theorem Th68: :: AFF_1:68
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 &
LIN b
2,b
3,b
5 &
LIN b
2,b
4,b
6 & b
3,b
4 // b
5,b
6 & b
5 = b
6 implies ( b
5 = b
2 & b
6 = b
2 ) )
theorem Th69: :: AFF_1:69
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 &
LIN b
2,b
3,b
5 &
LIN b
2,b
4,b
6 & b
3,b
4 // b
5,b
6 & b
5 = b
2 implies b
6 = b
2 )
theorem Th70: :: AFF_1:70
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 &
LIN b
2,b
3,b
5 &
LIN b
2,b
4,b
6 &
LIN b
2,b
4,b
7 & b
3,b
4 // b
5,b
6 & b
3,b
4 // b
5,b
7 implies b
6 = b
7 )
theorem Th71: :: AFF_1:71
theorem Th72: :: AFF_1:72
theorem Th73: :: AFF_1:73
theorem Th74: :: AFF_1:74
for b
1 being
AffinPlanefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( not b
2,b
3 // b
4,b
5 & ( for b
6 being
Element of b
1 holds
not (
LIN b
2,b
3,b
6 &
LIN b
4,b
5,b
6 ) ) )