:: AFF_4 semantic presentation
Lemma1:
for b1 being AffinSpace
for b2 being Subset of b1
for b3 being set holds
( b3 in b2 implies b3 is Element of b1 )
;
theorem Th1: :: AFF_4:1
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 or
LIN b
2,b
4,b
3 ) & b
2 <> b
3 & ( for b
6 being
Element of b
1 holds
not (
LIN b
2,b
5,b
6 & b
3,b
5 // b
4,b
6 ) ) )
theorem Th2: :: AFF_4:2
theorem Th3: :: AFF_4:3
for b
1 being
AffinSpacefor b
2, b
3 being
Element of b
1for b
4, b
5 being
Subset of b
1 holds
( ( b
2,b
3 // b
4 or b
3,b
2 // b
4 ) & b
4 // b
5 implies ( b
2,b
3 // b
5 & b
3,b
2 // b
5 ) )
theorem Th4: :: AFF_4:4
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 or b
3,b
2 // b
6 ) & ( b
2,b
3 // b
4,b
5 or b
4,b
5 // b
2,b
3 ) & b
2 <> b
3 implies ( b
4,b
5 // b
6 & b
5,b
4 // b
6 ) )
theorem Th5: :: AFF_4:5
for b
1 being
AffinSpacefor b
2, b
3 being
Element of b
1for b
4, b
5 being
Subset of b
1 holds
( ( b
2,b
3 // b
4 or b
3,b
2 // b
4 ) & ( b
2,b
3 // b
5 or b
3,b
2 // b
5 ) & b
2 <> b
3 implies b
4 // b
5 )
theorem Th6: :: AFF_4:6
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 or b
3,b
2 // b
6 ) & ( b
4,b
5 // b
6 or b
5,b
4 // b
6 ) implies b
2,b
3 // b
4,b
5 )
theorem Th7: :: AFF_4:7
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
6 // b
7 or b
7 // b
6 ) & b
2 <> b
3 & ( b
2,b
3 // b
4,b
5 or b
4,b
5 // b
2,b
3 ) & b
2 in b
6 & b
3 in b
6 & b
4 in b
7 implies b
5 in b
7 )
Lemma7:
for b1 being AffinSpace
for b2, b3, b4 being Element of b1
for b5, b6 being Subset of b1 holds
not ( b5 // b6 & b2 in b5 & b3 in b5 & b4 in b6 & ( for b7 being Element of b1 holds
not ( b7 in b6 & b2,b4 // b3,b7 ) ) )
theorem Th8: :: AFF_4:8
theorem Th9: :: AFF_4:9
theorem Th10: :: AFF_4:10
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
6 // b
7 or b
7 // b
6 ) & b
2 in b
6 & b
3 in b
6 & b
4 in b
7 & b
5 in b
7 & b
6 <> b
7 & ( b
2,b
4 // b
3,b
5 or b
4,b
2 // b
5,b
3 ) & b
2 = b
3 implies b
4 = b
5 )
theorem Th11: :: AFF_4:11
theorem Th12: :: AFF_4:12
:: deftheorem Def1 defines Plane AFF_4:def 1 :
:: deftheorem Def2 defines being_plane AFF_4:def 2 :
Lemma14:
for b1 being AffinSpace
for b2, b3 being Subset of b1
for b4 being Element of b1 holds
( b4 in Plane b2,b3 iff ex b5 being Element of b1 st
( b4,b5 // b2 & b5 in b3 ) )
theorem Th13: :: AFF_4:13
theorem Th14: :: AFF_4:14
theorem Th15: :: AFF_4:15
theorem Th16: :: AFF_4:16
theorem Th17: :: AFF_4:17
theorem Th18: :: AFF_4:18
Lemma18:
for b1 being AffinSpace
for b2 being Element of b1
for b3, b4, b5 being Subset of b1 holds
( b3 is_line & b4 is_line & b2 in b5 & b2 in Plane b3,b4 & b3 // b5 implies b5 c= Plane b3,b4 )
Lemma19:
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4, b5, b6 being Subset of b1 holds
( b4 is_line & b5 is_line & b6 is_line & b2 in Plane b4,b5 & b3 in Plane b4,b5 & b2 <> b3 & b2 in b6 & b3 in b6 implies b6 c= Plane b4,b5 )
theorem Th19: :: AFF_4:19
Lemma21:
for b1 being AffinSpace
for b2, b3, b4 being Subset of b1 holds
( b2 is_line & b3 is_line & b4 is_line & b4 c= Plane b2,b3 implies Plane b2,b4 c= Plane b2,b3 )
theorem Th20: :: AFF_4:20
theorem Th21: :: AFF_4:21
theorem Th22: :: AFF_4:22
theorem Th23: :: AFF_4:23
theorem Th24: :: AFF_4:24
theorem Th25: :: AFF_4:25
Lemma28:
for b1 being AffinSpace
for b2, b3, b4 being Element of b1
for b5 being Subset of b1 holds
not ( b5 is_line & b2 in b5 & b3 in b5 & b2 <> b3 & not b4 in b5 & LIN b2,b3,b4 )
theorem Th26: :: AFF_4:26
:: deftheorem Def3 defines * AFF_4:def 3 :
theorem Th27: :: AFF_4:27
theorem Th28: :: AFF_4:28
theorem Th29: :: AFF_4:29
theorem Th30: :: AFF_4:30
theorem Th31: :: AFF_4:31
Lemma35:
for b1 being AffinSpace
for b2 being Element of b1
for b3 being Subset of b1 holds
( b3 is_line & b2 in b3 implies b2 * b3 = b3 )
theorem Th32: :: AFF_4:32
:: deftheorem Def4 defines '||' AFF_4:def 4 :
theorem Th33: :: AFF_4:33
theorem Th34: :: AFF_4:34
Lemma40:
for b1 being AffinSpace
for b2 being Element of b1
for b3 being Subset of b1 holds
not ( b3 is_plane & ( for b4, b5 being Element of b1 holds
not ( b4 in b3 & b5 in b3 & not LIN b2,b4,b5 ) ) )
theorem Th35: :: AFF_4:35
theorem Th36: :: AFF_4:36
theorem Th37: :: AFF_4:37
theorem Th38: :: AFF_4:38
theorem Th39: :: AFF_4:39
theorem Th40: :: AFF_4:40
theorem Th41: :: AFF_4:41
theorem Th42: :: AFF_4:42
theorem Th43: :: AFF_4:43
:: deftheorem Def5 defines is_coplanar AFF_4:def 5 :
theorem Th44: :: AFF_4:44
for b
1 being
AffinSpacefor b
2, b
3, b
4 being
Subset of b
1 holds
( b
2,b
3,b
4 is_coplanar implies ( b
2,b
4,b
3 is_coplanar & b
3,b
2,b
4 is_coplanar & b
3,b
4,b
2 is_coplanar & b
4,b
2,b
3 is_coplanar & b
4,b
3,b
2 is_coplanar ) )
theorem Th45: :: AFF_4:45
theorem Th46: :: AFF_4:46
theorem Th47: :: AFF_4:47
theorem Th48: :: AFF_4:48
Lemma51:
for b1 being AffinSpace
for b2, b3, b4, b5, b6, b7, b8 being Element of b1
for b9, b10, b11 being Subset of b1 holds
( b2 in b9 & b2 in b10 & b2 in b11 & not b9,b10,b11 is_coplanar & b2 <> b3 & b2 <> b4 & b2 <> b5 & b3 in b9 & b6 in b9 & b4 in b10 & b7 in b10 & b5 in b11 & b8 in b11 & b9 is_line & b10 is_line & b11 is_line & b9 <> b10 & b9 <> b11 & b3,b4 // b6,b7 & b3,b5 // b6,b8 implies b4,b5 // b7,b8 )
theorem Th49: :: AFF_4:49
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1for b
9, b
10, b
11 being
Subset of b
1 holds
( not b
1 is
AffinPlane & b
2 in b
9 & b
2 in b
10 & b
2 in b
11 & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
3 in b
9 & b
6 in b
9 & b
4 in b
10 & b
7 in b
10 & b
5 in b
11 & b
8 in b
11 & b
9 is_line & b
10 is_line & b
11 is_line & b
9 <> b
10 & b
9 <> b
11 & b
3,b
4 // b
6,b
7 & b
3,b
5 // b
6,b
8 implies b
4,b
5 // b
7,b
8 )
theorem Th50: :: AFF_4:50
Lemma53:
for b1 being AffinSpace
for b2, b3, b4, b5, b6, b7 being Element of b1
for b8, b9, b10 being Subset of b1 holds
( b8 // b9 & b8 // b10 & not b8,b9,b10 is_coplanar & b2 in b8 & b3 in b8 & b4 in b9 & b5 in b9 & b6 in b10 & b7 in b10 & b8 is_line & b9 is_line & b10 is_line & b8 <> b9 & b8 <> b10 & b2,b4 // b3,b5 & b2,b6 // b3,b7 implies b4,b6 // b5,b7 )
theorem Th51: :: AFF_4:51
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1for b
8, b
9, b
10 being
Subset of b
1 holds
( not b
1 is
AffinPlane & b
8 // b
9 & b
8 // b
10 & b
2 in b
8 & b
3 in b
8 & b
4 in b
9 & b
5 in b
9 & b
6 in b
10 & b
7 in b
10 & b
8 is_line & b
9 is_line & b
10 is_line & b
8 <> b
9 & b
8 <> b
10 & b
2,b
4 // b
3,b
5 & b
2,b
6 // b
3,b
7 implies b
4,b
6 // b
5,b
7 )
theorem Th52: :: AFF_4:52
theorem Th53: :: AFF_4:53
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( b
1 is
AffinPlane & not
LIN b
2,b
3,b
4 & ( for b
7 being
Element of b
1 holds
not ( b
2,b
4 // b
5,b
7 & b
3,b
4 // b
6,b
7 ) ) )
Lemma56:
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being Element of b1
for b7 being Subset of b1 holds
not ( not LIN b2,b3,b4 & b5 <> b6 & b2,b3 // b5,b6 & b2 in b7 & b3 in b7 & b4 in b7 & b7 is_plane & b5 in b7 & ( for b8 being Element of b1 holds
not ( b2,b4 // b5,b8 & b3,b4 // b6,b8 ) ) )
theorem Th54: :: AFF_4:54
for b
1 being
AffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( not
LIN b
2,b
3,b
4 & b
5 <> b
6 & b
2,b
3 // b
5,b
6 & ( for b
7 being
Element of b
1 holds
not ( b
2,b
4 // b
5,b
7 & b
3,b
4 // b
6,b
7 ) ) )
theorem Th55: :: AFF_4:55
theorem Th56: :: AFF_4:56
theorem Th57: :: AFF_4:57
theorem Th58: :: AFF_4:58
theorem Th59: :: AFF_4:59
theorem Th60: :: AFF_4:60
Lemma63:
for b1 being AffinSpace
for b2, b3, b4 being Subset of b1 holds
( b2 is_plane & b3 is_plane & b4 is_plane & b2 '||' b3 & b3 '||' b4 implies b2 '||' b4 )
theorem Th61: :: AFF_4:61
Lemma65:
for b1 being AffinSpace
for b2, b3 being Subset of b1 holds
( b2 is_plane & b3 is_plane & b2 '||' b3 & b2 <> b3 implies for b4 being Element of b1 holds
not ( b4 in b2 & b4 in b3 ) )
theorem Th62: :: AFF_4:62
theorem Th63: :: AFF_4:63
theorem Th64: :: AFF_4:64
theorem Th65: :: AFF_4:65
:: deftheorem Def6 defines + AFF_4:def 6 :
theorem Th66: :: AFF_4:66
theorem Th67: :: AFF_4:67
theorem Th68: :: AFF_4:68
theorem Th69: :: AFF_4:69