:: INCSP_1 semantic presentation
:: deftheorem Def1 defines on INCSP_1:def 1 :
:: deftheorem Def2 defines on INCSP_1:def 2 :
:: deftheorem Def3 defines on INCSP_1:def 3 :
:: deftheorem Def4 defines on INCSP_1:def 4 :
:: deftheorem Def5 defines on INCSP_1:def 5 :
:: deftheorem Def6 defines linear INCSP_1:def 6 :
:: deftheorem Def7 defines planar INCSP_1:def 7 :
theorem Th1: :: INCSP_1:1
canceled;
theorem Th2: :: INCSP_1:2
canceled;
theorem Th3: :: INCSP_1:3
canceled;
theorem Th4: :: INCSP_1:4
canceled;
theorem Th5: :: INCSP_1:5
canceled;
theorem Th6: :: INCSP_1:6
canceled;
theorem Th7: :: INCSP_1:7
canceled;
theorem Th8: :: INCSP_1:8
canceled;
theorem Th9: :: INCSP_1:9
canceled;
theorem Th10: :: INCSP_1:10
canceled;
theorem Th11: :: INCSP_1:11
theorem Th12: :: INCSP_1:12
for b
1 being
IncStruct for b
2, b
3, b
4 being
POINT of b
1for b
5 being
LINE of b
1 holds
(
{b2,b3,b4} on b
5 iff ( b
2 on b
5 & b
3 on b
5 & b
4 on b
5 ) )
theorem Th13: :: INCSP_1:13
theorem Th14: :: INCSP_1:14
for b
1 being
IncStruct for b
2, b
3, b
4 being
POINT of b
1for b
5 being
PLANE of b
1 holds
(
{b2,b3,b4} on b
5 iff ( b
2 on b
5 & b
3 on b
5 & b
4 on b
5 ) )
theorem Th15: :: INCSP_1:15
for b
1 being
IncStruct for b
2, b
3, b
4, b
5 being
POINT of b
1for b
6 being
PLANE of b
1 holds
(
{b2,b3,b4,b5} on b
6 iff ( b
2 on b
6 & b
3 on b
6 & b
4 on b
6 & b
5 on b
6 ) )
theorem Th16: :: INCSP_1:16
theorem Th17: :: INCSP_1:17
theorem Th18: :: INCSP_1:18
theorem Th19: :: INCSP_1:19
theorem Th20: :: INCSP_1:20
theorem Th21: :: INCSP_1:21
theorem Th22: :: INCSP_1:22
theorem Th23: :: INCSP_1:23
definition
let c
1 be
IncStruct ;
attr a
1 is
IncSpace-like means :
Def8:
:: INCSP_1:def 8
( ( for b
1 being
LINE of a
1 holds
ex b
2, b
3 being
POINT of a
1 st
( b
2 <> b
3 &
{b2,b3} on b
1 ) ) & ( for b
1, b
2 being
POINT of a
1 holds
ex b
3 being
LINE of a
1 st
{b1,b2} on b
3 ) & ( for b
1, b
2 being
POINT of a
1for b
3, b
4 being
LINE of a
1 holds
( b
1 <> b
2 &
{b1,b2} on b
3 &
{b1,b2} on b
4 implies b
3 = b
4 ) ) & ( for b
1 being
PLANE of a
1 holds
ex b
2 being
POINT of a
1 st b
2 on b
1 ) & ( for b
1, b
2, b
3 being
POINT of a
1 holds
ex b
4 being
PLANE of a
1 st
{b1,b2,b3} on b
4 ) & ( for b
1, b
2, b
3 being
POINT of a
1for b
4, b
5 being
PLANE of a
1 holds
( not
{b1,b2,b3} is_collinear &
{b1,b2,b3} on b
4 &
{b1,b2,b3} on b
5 implies b
4 = b
5 ) ) & ( for b
1 being
LINE of a
1for b
2 being
PLANE of a
1 holds
( ex b
3, b
4 being
POINT of a
1 st
( b
3 <> b
4 &
{b3,b4} on b
1 &
{b3,b4} on b
2 ) implies b
1 on b
2 ) ) & ( for b
1 being
POINT of a
1for b
2, b
3 being
PLANE of a
1 holds
not ( b
1 on b
2 & b
1 on b
3 & ( for b
4 being
POINT of a
1 holds
not ( b
1 <> b
4 & b
4 on b
2 & b
4 on b
3 ) ) ) ) & not for b
1, b
2, b
3, b
4 being
POINT of a
1 holds
{b1,b2,b3,b4} is_coplanar & ( for b
1 being
POINT of a
1for b
2 being
LINE of a
1for b
3 being
PLANE of a
1 holds
( b
1 on b
2 & b
2 on b
3 implies b
1 on b
3 ) ) );
end;
:: deftheorem Def8 defines IncSpace-like INCSP_1:def 8 :
for b
1 being
IncStruct holds
( b
1 is
IncSpace-like iff ( ( for b
2 being
LINE of b
1 holds
ex b
3, b
4 being
POINT of b
1 st
( b
3 <> b
4 &
{b3,b4} on b
2 ) ) & ( for b
2, b
3 being
POINT of b
1 holds
ex b
4 being
LINE of b
1 st
{b2,b3} on b
4 ) & ( for b
2, b
3 being
POINT of b
1for b
4, b
5 being
LINE of b
1 holds
( b
2 <> b
3 &
{b2,b3} on b
4 &
{b2,b3} on b
5 implies b
4 = b
5 ) ) & ( for b
2 being
PLANE of b
1 holds
ex b
3 being
POINT of b
1 st b
3 on b
2 ) & ( for b
2, b
3, b
4 being
POINT of b
1 holds
ex b
5 being
PLANE of b
1 st
{b2,b3,b4} on b
5 ) & ( for b
2, b
3, b
4 being
POINT of b
1for b
5, b
6 being
PLANE of b
1 holds
( not
{b2,b3,b4} is_collinear &
{b2,b3,b4} on b
5 &
{b2,b3,b4} on b
6 implies b
5 = b
6 ) ) & ( for b
2 being
LINE of b
1for b
3 being
PLANE of b
1 holds
( ex b
4, b
5 being
POINT of b
1 st
( b
4 <> b
5 &
{b4,b5} on b
2 &
{b4,b5} on b
3 ) implies b
2 on b
3 ) ) & ( for b
2 being
POINT of b
1for b
3, b
4 being
PLANE of b
1 holds
not ( b
2 on b
3 & b
2 on b
4 & ( for b
5 being
POINT of b
1 holds
not ( b
2 <> b
5 & b
5 on b
3 & b
5 on b
4 ) ) ) ) & not for b
2, b
3, b
4, b
5 being
POINT of b
1 holds
{b2,b3,b4,b5} is_coplanar & ( for b
2 being
POINT of b
1for b
3 being
LINE of b
1for b
4 being
PLANE of b
1 holds
( b
2 on b
3 & b
3 on b
4 implies b
2 on b
4 ) ) ) );
theorem Th24: :: INCSP_1:24
canceled;
theorem Th25: :: INCSP_1:25
canceled;
theorem Th26: :: INCSP_1:26
canceled;
theorem Th27: :: INCSP_1:27
canceled;
theorem Th28: :: INCSP_1:28
canceled;
theorem Th29: :: INCSP_1:29
canceled;
theorem Th30: :: INCSP_1:30
canceled;
theorem Th31: :: INCSP_1:31
canceled;
theorem Th32: :: INCSP_1:32
canceled;
theorem Th33: :: INCSP_1:33
canceled;
theorem Th34: :: INCSP_1:34
canceled;
theorem Th35: :: INCSP_1:35
theorem Th36: :: INCSP_1:36
theorem Th37: :: INCSP_1:37
theorem Th38: :: INCSP_1:38
theorem Th39: :: INCSP_1:39
theorem Th40: :: INCSP_1:40
for b
1 being
IncSpacefor b
2, b
3, b
4, b
5 being
POINT of b
1for b
6 being
PLANE of b
1 holds
not ( not
{b2,b3,b4} is_collinear &
{b2,b3,b4} on b
6 & not b
5 on b
6 &
{b2,b3,b4,b5} is_coplanar )
theorem Th41: :: INCSP_1:41
for b
1 being
IncSpacefor b
2, b
3 being
LINE of b
1 holds
not ( ( for b
4 being
PLANE of b
1 holds
not ( b
2 on b
4 & b
3 on b
4 ) ) & not b
2 <> b
3 )
Lemma26:
for b1 being IncSpace
for b2 being POINT of b1
for b3 being LINE of b1 holds
ex b4 being POINT of b1 st
( b2 <> b4 & b4 on b3 )
theorem Th42: :: INCSP_1:42
for b
1 being
IncSpacefor b
2, b
3, b
4 being
LINE of b
1 holds
not ( ( for b
5 being
PLANE of b
1 holds
not ( b
2 on b
5 & b
3 on b
5 & b
4 on b
5 ) ) & ex b
5 being
POINT of b
1 st
( b
5 on b
2 & b
5 on b
3 & b
5 on b
4 ) & not b
2 <> b
3 )
theorem Th43: :: INCSP_1:43
for b
1 being
IncSpacefor b
2, b
3, b
4 being
LINE of b
1for b
5 being
PLANE of b
1 holds
( b
2 on b
5 & b
3 on b
5 & not b
4 on b
5 & b
2 <> b
3 implies for b
6 being
PLANE of b
1 holds
not ( b
4 on b
6 & b
2 on b
6 & b
3 on b
6 ) )
theorem Th44: :: INCSP_1:44
theorem Th45: :: INCSP_1:45
for b
1 being
IncSpacefor b
2, b
3 being
LINE of b
1 holds
not ( ex b
4 being
POINT of b
1 st
( b
4 on b
2 & b
4 on b
3 ) & ( for b
4 being
PLANE of b
1 holds
not ( b
2 on b
4 & b
3 on b
4 ) ) )
theorem Th46: :: INCSP_1:46
for b
1 being
IncSpacefor b
2, b
3 being
POINT of b
1 holds
not ( b
2 <> b
3 & ( for b
4 being
LINE of b
1 holds
not for b
5 being
LINE of b
1 holds
(
{b2,b3} on b
5 iff b
5 = b
4 ) ) )
theorem Th47: :: INCSP_1:47
theorem Th48: :: INCSP_1:48
for b
1 being
IncSpacefor b
2 being
POINT of b
1for b
3 being
LINE of b
1 holds
not ( not b
2 on b
3 & ( for b
4 being
PLANE of b
1 holds
not for b
5 being
PLANE of b
1 holds
( ( b
2 on b
5 & b
3 on b
5 ) iff b
4 = b
5 ) ) )
theorem Th49: :: INCSP_1:49
for b
1 being
IncSpacefor b
2, b
3 being
LINE of b
1 holds
not ( b
2 <> b
3 & ex b
4 being
POINT of b
1 st
( b
4 on b
2 & b
4 on b
3 ) & ( for b
4 being
PLANE of b
1 holds
not for b
5 being
PLANE of b
1 holds
( ( b
2 on b
5 & b
3 on b
5 ) iff b
4 = b
5 ) ) )
:: deftheorem Def9 defines Line INCSP_1:def 9 :
for b
1 being
IncSpacefor b
2, b
3 being
POINT of b
1 holds
( b
2 <> b
3 implies for b
4 being
LINE of b
1 holds
( b
4 = Line b
2,b
3 iff
{b2,b3} on b
4 ) );
definition
let c
1 be
IncSpace;
let c
2, c
3, c
4 be
POINT of c
1;
assume E32:
not
{c2,c3,c4} is_collinear
;
func Plane c
2,c
3,c
4 -> PLANE of a
1 means :
Def10:
:: INCSP_1:def 10
{a2,a3,a4} on a
5;
correctness
existence
ex b1 being PLANE of c1 st {c2,c3,c4} on b1;
uniqueness
for b1, b2 being PLANE of c1 holds
( {c2,c3,c4} on b1 & {c2,c3,c4} on b2 implies b1 = b2 );
by E32, Def8;
end;
:: deftheorem Def10 defines Plane INCSP_1:def 10 :
:: deftheorem Def11 defines Plane INCSP_1:def 11 :
for b
1 being
IncSpacefor b
2 being
POINT of b
1for b
3 being
LINE of b
1 holds
( not b
2 on b
3 implies for b
4 being
PLANE of b
1 holds
( b
4 = Plane b
2,b
3 iff ( b
2 on b
4 & b
3 on b
4 ) ) );
:: deftheorem Def12 defines Plane INCSP_1:def 12 :
for b
1 being
IncSpacefor b
2, b
3 being
LINE of b
1 holds
( b
2 <> b
3 & ex b
4 being
POINT of b
1 st
( b
4 on b
2 & b
4 on b
3 ) implies for b
4 being
PLANE of b
1 holds
( b
4 = Plane b
2,b
3 iff ( b
2 on b
4 & b
3 on b
4 ) ) );
theorem Th50: :: INCSP_1:50
canceled;
theorem Th51: :: INCSP_1:51
canceled;
theorem Th52: :: INCSP_1:52
canceled;
theorem Th53: :: INCSP_1:53
canceled;
theorem Th54: :: INCSP_1:54
canceled;
theorem Th55: :: INCSP_1:55
canceled;
theorem Th56: :: INCSP_1:56
canceled;
theorem Th57: :: INCSP_1:57
theorem Th58: :: INCSP_1:58
theorem Th59: :: INCSP_1:59
theorem Th60: :: INCSP_1:60
theorem Th61: :: INCSP_1:61
theorem Th62: :: INCSP_1:62
theorem Th63: :: INCSP_1:63
canceled;
theorem Th64: :: INCSP_1:64
theorem Th65: :: INCSP_1:65
theorem Th66: :: INCSP_1:66
theorem Th67: :: INCSP_1:67
theorem Th68: :: INCSP_1:68
for b
1 being
IncSpacefor b
2, b
3, b
4, b
5 being
POINT of b
1 holds
( not
{b2,b3,b4} is_collinear & b
5 on Plane b
2,b
3,b
4 implies
{b2,b3,b4,b5} is_coplanar )
theorem Th69: :: INCSP_1:69
theorem Th70: :: INCSP_1:70
Lemma39:
for b1 being IncSpace
for b2 being PLANE of b1 holds
ex b3, b4, b5, b6 being POINT of b1 st
( b3 on b2 & not {b3,b4,b5,b6} is_coplanar )
theorem Th71: :: INCSP_1:71
theorem Th72: :: INCSP_1:72
theorem Th73: :: INCSP_1:73
theorem Th74: :: INCSP_1:74
theorem Th75: :: INCSP_1:75
theorem Th76: :: INCSP_1:76
theorem Th77: :: INCSP_1:77
theorem Th78: :: INCSP_1:78
theorem Th79: :: INCSP_1:79
theorem Th80: :: INCSP_1:80
for b
1 being
IncSpacefor b
2 being
POINT of b
1for b
3 being
PLANE of b
1 holds
not ( b
2 on b
3 & ( for b
4, b
5, b
6 being
LINE of b
1 holds
not ( b
5 <> b
6 & b
5 on b
3 & b
6 on b
3 & not b
4 on b
3 & b
2 on b
4 & b
2 on b
5 & b
2 on b
6 ) ) )
theorem Th81: :: INCSP_1:81
for b
1 being
IncSpacefor b
2 being
POINT of b
1 holds
ex b
3, b
4, b
5 being
LINE of b
1 st
( b
2 on b
3 & b
2 on b
4 & b
2 on b
5 & ( for b
6 being
PLANE of b
1 holds
not ( b
3 on b
6 & b
4 on b
6 & b
5 on b
6 ) ) )
theorem Th82: :: INCSP_1:82
theorem Th83: :: INCSP_1:83
theorem Th84: :: INCSP_1:84
for b
1 being
IncSpacefor b
2 being
LINE of b
1 holds
ex b
3 being
LINE of b
1 st
for b
4 being
PLANE of b
1 holds
not ( b
2 on b
4 & b
3 on b
4 )
theorem Th85: :: INCSP_1:85
for b
1 being
IncSpacefor b
2 being
LINE of b
1 holds
ex b
3, b
4 being
PLANE of b
1 st
( b
3 <> b
4 & b
2 on b
3 & b
2 on b
4 )
theorem Th86: :: INCSP_1:86
canceled;
theorem Th87: :: INCSP_1:87
theorem Th88: :: INCSP_1:88
for b
1 being
IncSpacefor b
2, b
3 being
PLANE of b
1 holds
not ( b
2 <> b
3 & ex b
4 being
POINT of b
1 st
( b
4 on b
2 & b
4 on b
3 ) & ( for b
4 being
LINE of b
1 holds
not for b
5 being
POINT of b
1 holds
( ( b
5 on b
2 & b
5 on b
3 ) iff b
5 on b
4 ) ) )