:: INCSP_1 semantic presentation

definition
attr a1 is strict;
struct IncProjStr -> ;
aggr IncProjStr(# Points, Lines, Inc #) -> IncProjStr ;
sel Points c1 -> non empty set ;
sel Lines c1 -> non empty set ;
sel Inc c1 -> Relation of the Points of a1,the Lines of a1;
end;

definition
attr a1 is strict;
struct IncStruct -> IncProjStr ;
aggr IncStruct(# Points, Lines, Planes, Inc, Inc2, Inc3 #) -> IncStruct ;
sel Planes c1 -> non empty set ;
sel Inc2 c1 -> Relation of the Points of a1,the Planes of a1;
sel Inc3 c1 -> Relation of the Lines of a1,the Planes of a1;
end;

definition
let c1 be IncProjStr ;
mode POINT is Element of the Points of a1;
mode LINE is Element of the Lines of a1;
end;

definition
let c1 be IncStruct ;
mode PLANE is Element of the Planes of a1;
end;

definition
let c1 be IncProjStr ;
let c2 be POINT of c1;
let c3 be LINE of c1;
pred c2 on c3 means :Def1: :: INCSP_1:def 1
[a2,a3] in the Inc of a1;
end;

:: deftheorem Def1 defines on INCSP_1:def 1 :
for b1 being IncProjStr
for b2 being POINT of b1
for b3 being LINE of b1 holds
( b2 on b3 iff [b2,b3] in the Inc of b1 );

definition
let c1 be IncStruct ;
let c2 be POINT of c1;
let c3 be PLANE of c1;
pred c2 on c3 means :Def2: :: INCSP_1:def 2
[a2,a3] in the Inc2 of a1;
end;

:: deftheorem Def2 defines on INCSP_1:def 2 :
for b1 being IncStruct
for b2 being POINT of b1
for b3 being PLANE of b1 holds
( b2 on b3 iff [b2,b3] in the Inc2 of b1 );

definition
let c1 be IncStruct ;
let c2 be LINE of c1;
let c3 be PLANE of c1;
pred c2 on c3 means :Def3: :: INCSP_1:def 3
[a2,a3] in the Inc3 of a1;
end;

:: deftheorem Def3 defines on INCSP_1:def 3 :
for b1 being IncStruct
for b2 being LINE of b1
for b3 being PLANE of b1 holds
( b2 on b3 iff [b2,b3] in the Inc3 of b1 );

definition
let c1 be IncProjStr ;
let c2 be Subset of the Points of c1;
let c3 be LINE of c1;
pred c2 on c3 means :Def4: :: INCSP_1:def 4
for b1 being POINT of a1 holds
( b1 in a2 implies b1 on a3 );
end;

:: deftheorem Def4 defines on INCSP_1:def 4 :
for b1 being IncProjStr
for b2 being Subset of the Points of b1
for b3 being LINE of b1 holds
( b2 on b3 iff for b4 being POINT of b1 holds
( b4 in b2 implies b4 on b3 ) );

definition
let c1 be IncStruct ;
let c2 be Subset of the Points of c1;
let c3 be PLANE of c1;
pred c2 on c3 means :Def5: :: INCSP_1:def 5
for b1 being POINT of a1 holds
( b1 in a2 implies b1 on a3 );
end;

:: deftheorem Def5 defines on INCSP_1:def 5 :
for b1 being IncStruct
for b2 being Subset of the Points of b1
for b3 being PLANE of b1 holds
( b2 on b3 iff for b4 being POINT of b1 holds
( b4 in b2 implies b4 on b3 ) );

definition
let c1 be IncProjStr ;
let c2 be Subset of the Points of c1;
attr a2 is linear means :Def6: :: INCSP_1:def 6
ex b1 being LINE of a1 st a2 on b1;
end;

:: deftheorem Def6 defines linear INCSP_1:def 6 :
for b1 being IncProjStr
for b2 being Subset of the Points of b1 holds
( b2 is linear iff ex b3 being LINE of b1 st b2 on b3 );

notation
let c1 be IncProjStr ;
let c2 be Subset of the Points of c1;
synonym c2 is_collinear for linear c2;
end;

definition
let c1 be IncStruct ;
let c2 be Subset of the Points of c1;
attr a2 is planar means :Def7: :: INCSP_1:def 7
ex b1 being PLANE of a1 st a2 on b1;
end;

:: deftheorem Def7 defines planar INCSP_1:def 7 :
for b1 being IncStruct
for b2 being Subset of the Points of b1 holds
( b2 is planar iff ex b3 being PLANE of b1 st b2 on b3 );

notation
let c1 be IncStruct ;
let c2 be Subset of the Points of c1;
synonym c2 is_coplanar for planar c2;
end;

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
for b1 being IncStruct
for b2, b3 being POINT of b1
for b4 being LINE of b1 holds
( {b2,b3} on b4 iff ( b2 on b4 & b3 on b4 ) )
proof end;

theorem Th12: :: INCSP_1:12
for b1 being IncStruct
for b2, b3, b4 being POINT of b1
for b5 being LINE of b1 holds
( {b2,b3,b4} on b5 iff ( b2 on b5 & b3 on b5 & b4 on b5 ) )
proof end;

theorem Th13: :: INCSP_1:13
for b1 being IncStruct
for b2, b3 being POINT of b1
for b4 being PLANE of b1 holds
( {b2,b3} on b4 iff ( b2 on b4 & b3 on b4 ) )
proof end;

theorem Th14: :: INCSP_1:14
for b1 being IncStruct
for b2, b3, b4 being POINT of b1
for b5 being PLANE of b1 holds
( {b2,b3,b4} on b5 iff ( b2 on b5 & b3 on b5 & b4 on b5 ) )
proof end;

theorem Th15: :: INCSP_1:15
for b1 being IncStruct
for b2, b3, b4, b5 being POINT of b1
for b6 being PLANE of b1 holds
( {b2,b3,b4,b5} on b6 iff ( b2 on b6 & b3 on b6 & b4 on b6 & b5 on b6 ) )
proof end;

theorem Th16: :: INCSP_1:16
for b1 being IncStruct
for b2 being LINE of b1
for b3, b4 being Subset of the Points of b1 holds
( b3 c= b4 & b4 on b2 implies b3 on b2 )
proof end;

theorem Th17: :: INCSP_1:17
for b1 being IncStruct
for b2 being PLANE of b1
for b3, b4 being Subset of the Points of b1 holds
( b3 c= b4 & b4 on b2 implies b3 on b2 )
proof end;

theorem Th18: :: INCSP_1:18
for b1 being IncStruct
for b2 being POINT of b1
for b3 being LINE of b1
for b4 being Subset of the Points of b1 holds
( ( b4 on b3 & b2 on b3 ) iff b4 \/ {b2} on b3 )
proof end;

theorem Th19: :: INCSP_1:19
for b1 being IncStruct
for b2 being POINT of b1
for b3 being PLANE of b1
for b4 being Subset of the Points of b1 holds
( ( b4 on b3 & b2 on b3 ) iff b4 \/ {b2} on b3 )
proof end;

theorem Th20: :: INCSP_1:20
for b1 being IncStruct
for b2 being LINE of b1
for b3, b4 being Subset of the Points of b1 holds
( b3 \/ b4 on b2 iff ( b3 on b2 & b4 on b2 ) )
proof end;

theorem Th21: :: INCSP_1:21
for b1 being IncStruct
for b2 being PLANE of b1
for b3, b4 being Subset of the Points of b1 holds
( b3 \/ b4 on b2 iff ( b3 on b2 & b4 on b2 ) )
proof end;

theorem Th22: :: INCSP_1:22
for b1 being IncStruct
for b2, b3 being Subset of the Points of b1 holds
( b2 c= b3 & b3 is_collinear implies b2 is_collinear )
proof end;

theorem Th23: :: INCSP_1:23
for b1 being IncStruct
for b2, b3 being Subset of the Points of b1 holds
( b2 c= b3 & b3 is_coplanar implies b2 is_coplanar )
proof end;

definition
let c1 be IncStruct ;
attr a1 is IncSpace-like means :Def8: :: INCSP_1:def 8
( ( for b1 being LINE of a1 holds
ex b2, b3 being POINT of a1 st
( b2 <> b3 & {b2,b3} on b1 ) ) & ( for b1, b2 being POINT of a1 holds
ex b3 being LINE of a1 st {b1,b2} on b3 ) & ( for b1, b2 being POINT of a1
for b3, b4 being LINE of a1 holds
( b1 <> b2 & {b1,b2} on b3 & {b1,b2} on b4 implies b3 = b4 ) ) & ( for b1 being PLANE of a1 holds
ex b2 being POINT of a1 st b2 on b1 ) & ( for b1, b2, b3 being POINT of a1 holds
ex b4 being PLANE of a1 st {b1,b2,b3} on b4 ) & ( for b1, b2, b3 being POINT of a1
for b4, b5 being PLANE of a1 holds
( not {b1,b2,b3} is_collinear & {b1,b2,b3} on b4 & {b1,b2,b3} on b5 implies b4 = b5 ) ) & ( for b1 being LINE of a1
for b2 being PLANE of a1 holds
( ex b3, b4 being POINT of a1 st
( b3 <> b4 & {b3,b4} on b1 & {b3,b4} on b2 ) implies b1 on b2 ) ) & ( for b1 being POINT of a1
for b2, b3 being PLANE of a1 holds
not ( b1 on b2 & b1 on b3 & ( for b4 being POINT of a1 holds
not ( b1 <> b4 & b4 on b2 & b4 on b3 ) ) ) ) & not for b1, b2, b3, b4 being POINT of a1 holds {b1,b2,b3,b4} is_coplanar & ( for b1 being POINT of a1
for b2 being LINE of a1
for b3 being PLANE of a1 holds
( b1 on b2 & b2 on b3 implies b1 on b3 ) ) );
end;

:: deftheorem Def8 defines IncSpace-like INCSP_1:def 8 :
for b1 being IncStruct holds
( b1 is IncSpace-like iff ( ( for b2 being LINE of b1 holds
ex b3, b4 being POINT of b1 st
( b3 <> b4 & {b3,b4} on b2 ) ) & ( for b2, b3 being POINT of b1 holds
ex b4 being LINE of b1 st {b2,b3} on b4 ) & ( for b2, b3 being POINT of b1
for b4, b5 being LINE of b1 holds
( b2 <> b3 & {b2,b3} on b4 & {b2,b3} on b5 implies b4 = b5 ) ) & ( for b2 being PLANE of b1 holds
ex b3 being POINT of b1 st b3 on b2 ) & ( for b2, b3, b4 being POINT of b1 holds
ex b5 being PLANE of b1 st {b2,b3,b4} on b5 ) & ( for b2, b3, b4 being POINT of b1
for b5, b6 being PLANE of b1 holds
( not {b2,b3,b4} is_collinear & {b2,b3,b4} on b5 & {b2,b3,b4} on b6 implies b5 = b6 ) ) & ( for b2 being LINE of b1
for b3 being PLANE of b1 holds
( ex b4, b5 being POINT of b1 st
( b4 <> b5 & {b4,b5} on b2 & {b4,b5} on b3 ) implies b2 on b3 ) ) & ( for b2 being POINT of b1
for b3, b4 being PLANE of b1 holds
not ( b2 on b3 & b2 on b4 & ( for b5 being POINT of b1 holds
not ( b2 <> b5 & b5 on b3 & b5 on b4 ) ) ) ) & not for b2, b3, b4, b5 being POINT of b1 holds {b2,b3,b4,b5} is_coplanar & ( for b2 being POINT of b1
for b3 being LINE of b1
for b4 being PLANE of b1 holds
( b2 on b3 & b3 on b4 implies b2 on b4 ) ) ) );

registration
cluster strict IncSpace-like IncStruct ;
existence
ex b1 being IncStruct st
( b1 is strict & b1 is IncSpace-like )
proof end;
end;

definition
mode IncSpace is IncSpace-like IncStruct ;
end;

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
for b1 being IncSpace
for b2 being LINE of b1
for b3 being PLANE of b1
for b4 being Subset of the Points of b1 holds
( b4 on b2 & b2 on b3 implies b4 on b3 )
proof end;

theorem Th36: :: INCSP_1:36
for b1 being IncSpace
for b2, b3 being POINT of b1 holds {b2,b2,b3} is_collinear
proof end;

theorem Th37: :: INCSP_1:37
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds {b2,b2,b3,b4} is_coplanar
proof end;

theorem Th38: :: INCSP_1:38
for b1 being IncSpace
for b2, b3, b4, b5 being POINT of b1 holds
( {b2,b3,b4} is_collinear implies {b2,b3,b4,b5} is_coplanar )
proof end;

theorem Th39: :: INCSP_1:39
for b1 being IncSpace
for b2, b3, b4 being POINT of b1
for b5 being LINE of b1 holds
not ( b2 <> b3 & {b2,b3} on b5 & not b4 on b5 & {b2,b3,b4} is_collinear )
proof end;

theorem Th40: :: INCSP_1:40
for b1 being IncSpace
for b2, b3, b4, b5 being POINT of b1
for b6 being PLANE of b1 holds
not ( not {b2,b3,b4} is_collinear & {b2,b3,b4} on b6 & not b5 on b6 & {b2,b3,b4,b5} is_coplanar )
proof end;

theorem Th41: :: INCSP_1:41
for b1 being IncSpace
for b2, b3 being LINE of b1 holds
not ( ( for b4 being PLANE of b1 holds
not ( b2 on b4 & b3 on b4 ) ) & not b2 <> b3 )
proof end;

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 )
proof end;

theorem Th42: :: INCSP_1:42
for b1 being IncSpace
for b2, b3, b4 being LINE of b1 holds
not ( ( for b5 being PLANE of b1 holds
not ( b2 on b5 & b3 on b5 & b4 on b5 ) ) & ex b5 being POINT of b1 st
( b5 on b2 & b5 on b3 & b5 on b4 ) & not b2 <> b3 )
proof end;

theorem Th43: :: INCSP_1:43
for b1 being IncSpace
for b2, b3, b4 being LINE of b1
for b5 being PLANE of b1 holds
( b2 on b5 & b3 on b5 & not b4 on b5 & b2 <> b3 implies for b6 being PLANE of b1 holds
not ( b4 on b6 & b2 on b6 & b3 on b6 ) )
proof end;

theorem Th44: :: INCSP_1:44
for b1 being IncSpace
for b2 being POINT of b1
for b3 being LINE of b1 holds
ex b4 being PLANE of b1 st
( b2 on b4 & b3 on b4 )
proof end;

theorem Th45: :: INCSP_1:45
for b1 being IncSpace
for b2, b3 being LINE of b1 holds
not ( ex b4 being POINT of b1 st
( b4 on b2 & b4 on b3 ) & ( for b4 being PLANE of b1 holds
not ( b2 on b4 & b3 on b4 ) ) )
proof end;

theorem Th46: :: INCSP_1:46
for b1 being IncSpace
for b2, b3 being POINT of b1 holds
not ( b2 <> b3 & ( for b4 being LINE of b1 holds
not for b5 being LINE of b1 holds
( {b2,b3} on b5 iff b5 = b4 ) ) )
proof end;

theorem Th47: :: INCSP_1:47
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
not ( not {b2,b3,b4} is_collinear & ( for b5 being PLANE of b1 holds
not for b6 being PLANE of b1 holds
( {b2,b3,b4} on b6 iff b5 = b6 ) ) )
proof end;

theorem Th48: :: INCSP_1:48
for b1 being IncSpace
for b2 being POINT of b1
for b3 being LINE of b1 holds
not ( not b2 on b3 & ( for b4 being PLANE of b1 holds
not for b5 being PLANE of b1 holds
( ( b2 on b5 & b3 on b5 ) iff b4 = b5 ) ) )
proof end;

theorem Th49: :: INCSP_1:49
for b1 being IncSpace
for b2, b3 being LINE of b1 holds
not ( b2 <> b3 & ex b4 being POINT of b1 st
( b4 on b2 & b4 on b3 ) & ( for b4 being PLANE of b1 holds
not for b5 being PLANE of b1 holds
( ( b2 on b5 & b3 on b5 ) iff b4 = b5 ) ) )
proof end;

definition
let c1 be IncSpace;
let c2, c3 be POINT of c1;
assume E31: c2 <> c3 ;
func Line c2,c3 -> LINE of a1 means :Def9: :: INCSP_1:def 9
{a2,a3} on a4;
correctness
existence
ex b1 being LINE of c1 st {c2,c3} on b1
;
uniqueness
for b1, b2 being LINE of c1 holds
( {c2,c3} on b1 & {c2,c3} on b2 implies b1 = b2 )
;
by E31, Def8;
end;

:: deftheorem Def9 defines Line INCSP_1:def 9 :
for b1 being IncSpace
for b2, b3 being POINT of b1 holds
( b2 <> b3 implies for b4 being LINE of b1 holds
( b4 = Line b2,b3 iff {b2,b3} on b4 ) );

definition
let c1 be IncSpace;
let c2, c3, c4 be POINT of c1;
assume E32: not {c2,c3,c4} is_collinear ;
func Plane c2,c3,c4 -> PLANE of a1 means :Def10: :: INCSP_1:def 10
{a2,a3,a4} on a5;
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 :
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
( not {b2,b3,b4} is_collinear implies for b5 being PLANE of b1 holds
( b5 = Plane b2,b3,b4 iff {b2,b3,b4} on b5 ) );

definition
let c1 be IncSpace;
let c2 be POINT of c1;
let c3 be LINE of c1;
assume E33: not c2 on c3 ;
func Plane c2,c3 -> PLANE of a1 means :Def11: :: INCSP_1:def 11
( a2 on a4 & a3 on a4 );
existence
ex b1 being PLANE of c1 st
( c2 on b1 & c3 on b1 )
by Th44;
uniqueness
for b1, b2 being PLANE of c1 holds
( c2 on b1 & c3 on b1 & c2 on b2 & c3 on b2 implies b1 = b2 )
proof end;
end;

:: deftheorem Def11 defines Plane INCSP_1:def 11 :
for b1 being IncSpace
for b2 being POINT of b1
for b3 being LINE of b1 holds
( not b2 on b3 implies for b4 being PLANE of b1 holds
( b4 = Plane b2,b3 iff ( b2 on b4 & b3 on b4 ) ) );

definition
let c1 be IncSpace;
let c2, c3 be LINE of c1;
assume that
E34: c2 <> c3 and
E35: ex b1 being POINT of c1 st
( b1 on c2 & b1 on c3 ) ;
func Plane c2,c3 -> PLANE of a1 means :Def12: :: INCSP_1:def 12
( a2 on a4 & a3 on a4 );
existence
ex b1 being PLANE of c1 st
( c2 on b1 & c3 on b1 )
by E35, Th45;
uniqueness
for b1, b2 being PLANE of c1 holds
( c2 on b1 & c3 on b1 & c2 on b2 & c3 on b2 implies b1 = b2 )
proof end;
end;

:: deftheorem Def12 defines Plane INCSP_1:def 12 :
for b1 being IncSpace
for b2, b3 being LINE of b1 holds
( b2 <> b3 & ex b4 being POINT of b1 st
( b4 on b2 & b4 on b3 ) implies for b4 being PLANE of b1 holds
( b4 = Plane b2,b3 iff ( b2 on b4 & b3 on b4 ) ) );

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
for b1 being IncSpace
for b2, b3 being POINT of b1 holds
( b2 <> b3 implies Line b2,b3 = Line b3,b2 )
proof end;

theorem Th58: :: INCSP_1:58
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
( not {b2,b3,b4} is_collinear implies Plane b2,b3,b4 = Plane b2,b4,b3 )
proof end;

theorem Th59: :: INCSP_1:59
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
( not {b2,b3,b4} is_collinear implies Plane b2,b3,b4 = Plane b3,b2,b4 )
proof end;

theorem Th60: :: INCSP_1:60
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
( not {b2,b3,b4} is_collinear implies Plane b2,b3,b4 = Plane b3,b4,b2 )
proof end;

theorem Th61: :: INCSP_1:61
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
( not {b2,b3,b4} is_collinear implies Plane b2,b3,b4 = Plane b4,b2,b3 )
proof end;

theorem Th62: :: INCSP_1:62
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
( not {b2,b3,b4} is_collinear implies Plane b2,b3,b4 = Plane b4,b3,b2 )
proof end;

theorem Th63: :: INCSP_1:63
canceled;

theorem Th64: :: INCSP_1:64
for b1 being IncSpace
for b2, b3 being LINE of b1 holds
( b2 <> b3 & ex b4 being POINT of b1 st
( b4 on b2 & b4 on b3 ) implies Plane b2,b3 = Plane b3,b2 )
proof end;

theorem Th65: :: INCSP_1:65
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
( b2 <> b3 & b4 on Line b2,b3 implies {b2,b3,b4} is_collinear )
proof end;

theorem Th66: :: INCSP_1:66
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
( b2 <> b3 & b2 <> b4 & {b2,b3,b4} is_collinear implies Line b2,b3 = Line b2,b4 )
proof end;

theorem Th67: :: INCSP_1:67
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
( not {b2,b3,b4} is_collinear implies Plane b2,b3,b4 = Plane b4,(Line b2,b3) )
proof end;

theorem Th68: :: INCSP_1:68
for b1 being IncSpace
for b2, b3, b4, b5 being POINT of b1 holds
( not {b2,b3,b4} is_collinear & b5 on Plane b2,b3,b4 implies {b2,b3,b4,b5} is_coplanar )
proof end;

theorem Th69: :: INCSP_1:69
for b1 being IncSpace
for b2, b3, b4 being POINT of b1
for b5 being LINE of b1 holds
( not b2 on b5 & {b3,b4} on b5 & b3 <> b4 implies Plane b2,b5 = Plane b3,b4,b2 )
proof end;

theorem Th70: :: INCSP_1:70
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
( not {b2,b3,b4} is_collinear implies Plane b2,b3,b4 = Plane (Line b2,b3),(Line b2,b4) )
proof end;

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 )
proof end;

theorem Th71: :: INCSP_1:71
for b1 being IncSpace
for b2 being PLANE of b1 holds
ex b3, b4, b5 being POINT of b1 st
( {b3,b4,b5} on b2 & not {b3,b4,b5} is_collinear )
proof end;

theorem Th72: :: INCSP_1:72
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 ) by Lemma39;

theorem Th73: :: INCSP_1:73
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 ) by Lemma26;

theorem Th74: :: INCSP_1:74
for b1 being IncSpace
for b2, b3 being POINT of b1
for b4 being PLANE of b1 holds
not ( b2 <> b3 & ( for b5 being POINT of b1 holds
not ( b5 on b4 & not {b2,b3,b5} is_collinear ) ) )
proof end;

theorem Th75: :: INCSP_1:75
for b1 being IncSpace
for b2, b3, b4 being POINT of b1 holds
not ( not {b2,b3,b4} is_collinear & ( for b5 being POINT of b1 holds {b2,b3,b4,b5} is_coplanar ) )
proof end;

theorem Th76: :: INCSP_1:76
for b1 being IncSpace
for b2 being POINT of b1
for b3 being PLANE of b1 holds
ex b4, b5 being POINT of b1 st
( {b4,b5} on b3 & not {b2,b4,b5} is_collinear )
proof end;

theorem Th77: :: INCSP_1:77
for b1 being IncSpace
for b2, b3 being POINT of b1 holds
not ( b2 <> b3 & ( for b4, b5 being POINT of b1 holds {b2,b3,b4,b5} is_coplanar ) )
proof end;

theorem Th78: :: INCSP_1:78
for b1 being IncSpace
for b2 being POINT of b1 holds
not for b3, b4, b5 being POINT of b1 holds {b2,b3,b4,b5} is_coplanar
proof end;

theorem Th79: :: INCSP_1:79
for b1 being IncSpace
for b2 being POINT of b1
for b3 being PLANE of b1 holds
ex b4 being LINE of b1 st
( not b2 on b4 & b4 on b3 )
proof end;

theorem Th80: :: INCSP_1:80
for b1 being IncSpace
for b2 being POINT of b1
for b3 being PLANE of b1 holds
not ( b2 on b3 & ( for b4, b5, b6 being LINE of b1 holds
not ( b5 <> b6 & b5 on b3 & b6 on b3 & not b4 on b3 & b2 on b4 & b2 on b5 & b2 on b6 ) ) )
proof end;

theorem Th81: :: INCSP_1:81
for b1 being IncSpace
for b2 being POINT of b1 holds
ex b3, b4, b5 being LINE of b1 st
( b2 on b3 & b2 on b4 & b2 on b5 & ( for b6 being PLANE of b1 holds
not ( b3 on b6 & b4 on b6 & b5 on b6 ) ) )
proof end;

theorem Th82: :: INCSP_1:82
for b1 being IncSpace
for b2 being POINT of b1
for b3 being LINE of b1 holds
ex b4 being PLANE of b1 st
( b2 on b4 & not b3 on b4 )
proof end;

theorem Th83: :: INCSP_1:83
for b1 being IncSpace
for b2 being LINE of b1
for b3 being PLANE of b1 holds
ex b4 being POINT of b1 st
( b4 on b3 & not b4 on b2 )
proof end;

theorem Th84: :: INCSP_1:84
for b1 being IncSpace
for b2 being LINE of b1 holds
ex b3 being LINE of b1 st
for b4 being PLANE of b1 holds
not ( b2 on b4 & b3 on b4 )
proof end;

theorem Th85: :: INCSP_1:85
for b1 being IncSpace
for b2 being LINE of b1 holds
ex b3, b4 being PLANE of b1 st
( b3 <> b4 & b2 on b3 & b2 on b4 )
proof end;

theorem Th86: :: INCSP_1:86
canceled;

theorem Th87: :: INCSP_1:87
for b1 being IncSpace
for b2, b3 being POINT of b1
for b4 being LINE of b1
for b5 being PLANE of b1 holds
( not b4 on b5 & {b2,b3} on b4 & {b2,b3} on b5 implies b2 = b3 ) by Def8;

theorem Th88: :: INCSP_1:88
for b1 being IncSpace
for b2, b3 being PLANE of b1 holds
not ( b2 <> b3 & ex b4 being POINT of b1 st
( b4 on b2 & b4 on b3 ) & ( for b4 being LINE of b1 holds
not for b5 being POINT of b1 holds
( ( b5 on b2 & b5 on b3 ) iff b5 on b4 ) ) )
proof end;