:: 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 S be IncProjStr ;
mode POINT of a1 is Element of the Points of a1;
mode LINE of a1 is Element of the Lines of a1;
end;

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

definition
let S be IncProjStr ;
let A be POINT of S;
let L be LINE of S;
pred c2 on c3 means :Def1: :: INCSP_1:def 1
[A,L] in the Inc of S;
end;

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

definition
let S be IncStruct ;
let A be POINT of S;
let P be PLANE of S;
pred c2 on c3 means :Def2: :: INCSP_1:def 2
[A,P] in the Inc2 of S;
end;

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

definition
let S be IncStruct ;
let L be LINE of S;
let P be PLANE of S;
pred c2 on c3 means :Def3: :: INCSP_1:def 3
[L,P] in the Inc3 of S;
end;

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

definition
let S be IncProjStr ;
let F be Subset of the Points of S;
let L be LINE of S;
pred c2 on c3 means :Def4: :: INCSP_1:def 4
for A being POINT of S st A in F holds
A on L;
end;

:: deftheorem Def4 defines on INCSP_1:def 4 :
for S being IncProjStr
for F being Subset of the Points of S
for L being LINE of S holds
( F on L iff for A being POINT of S st A in F holds
A on L );

definition
let S be IncStruct ;
let F be Subset of the Points of S;
let P be PLANE of S;
pred c2 on c3 means :Def5: :: INCSP_1:def 5
for A being POINT of S st A in F holds
A on P;
end;

:: deftheorem Def5 defines on INCSP_1:def 5 :
for S being IncStruct
for F being Subset of the Points of S
for P being PLANE of S holds
( F on P iff for A being POINT of S st A in F holds
A on P );

definition
let S be IncProjStr ;
let F be Subset of the Points of S;
attr a2 is linear means :Def6: :: INCSP_1:def 6
ex L being LINE of S st F on L;
end;

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

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

definition
let S be IncStruct ;
let F be Subset of the Points of S;
attr a2 is planar means :Def7: :: INCSP_1:def 7
ex P being PLANE of S st F on P;
end;

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

notation
let S be IncStruct ;
let F be Subset of the Points of S;
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 S being IncStruct
for A, B being POINT of S
for L being LINE of S holds
( {A,B} on L iff ( A on L & B on L ) )
proof end;

theorem Th12: :: INCSP_1:12
for S being IncStruct
for A, B, C being POINT of S
for L being LINE of S holds
( {A,B,C} on L iff ( A on L & B on L & C on L ) )
proof end;

theorem Th13: :: INCSP_1:13
for S being IncStruct
for A, B being POINT of S
for P being PLANE of S holds
( {A,B} on P iff ( A on P & B on P ) )
proof end;

theorem Th14: :: INCSP_1:14
for S being IncStruct
for A, B, C being POINT of S
for P being PLANE of S holds
( {A,B,C} on P iff ( A on P & B on P & C on P ) )
proof end;

theorem Th15: :: INCSP_1:15
for S being IncStruct
for A, B, C, D being POINT of S
for P being PLANE of S holds
( {A,B,C,D} on P iff ( A on P & B on P & C on P & D on P ) )
proof end;

theorem Th16: :: INCSP_1:16
for S being IncStruct
for L being LINE of S
for G, F being Subset of the Points of S st G c= F & F on L holds
G on L
proof end;

theorem Th17: :: INCSP_1:17
for S being IncStruct
for P being PLANE of S
for G, F being Subset of the Points of S st G c= F & F on P holds
G on P
proof end;

theorem Th18: :: INCSP_1:18
for S being IncStruct
for A being POINT of S
for L being LINE of S
for F being Subset of the Points of S holds
( ( F on L & A on L ) iff F \/ {A} on L )
proof end;

theorem Th19: :: INCSP_1:19
for S being IncStruct
for A being POINT of S
for P being PLANE of S
for F being Subset of the Points of S holds
( ( F on P & A on P ) iff F \/ {A} on P )
proof end;

theorem Th20: :: INCSP_1:20
for S being IncStruct
for L being LINE of S
for F, G being Subset of the Points of S holds
( F \/ G on L iff ( F on L & G on L ) )
proof end;

theorem Th21: :: INCSP_1:21
for S being IncStruct
for P being PLANE of S
for F, G being Subset of the Points of S holds
( F \/ G on P iff ( F on P & G on P ) )
proof end;

theorem Th22: :: INCSP_1:22
for S being IncStruct
for G, F being Subset of the Points of S st G c= F & F is_collinear holds
G is_collinear
proof end;

theorem Th23: :: INCSP_1:23
for S being IncStruct
for G, F being Subset of the Points of S st G c= F & F is_coplanar holds
G is_coplanar
proof end;

definition
let IT be IncStruct ;
attr a1 is IncSpace-like means :Def8: :: INCSP_1:def 8
( ( for L being LINE of IT ex A, B being POINT of IT st
( A <> B & {A,B} on L ) ) & ( for A, B being POINT of IT ex L being LINE of IT st {A,B} on L ) & ( for A, B being POINT of IT
for K, L being LINE of IT st A <> B & {A,B} on K & {A,B} on L holds
K = L ) & ( for P being PLANE of IT ex A being POINT of IT st A on P ) & ( for A, B, C being POINT of IT ex P being PLANE of IT st {A,B,C} on P ) & ( for A, B, C being POINT of IT
for P, Q being PLANE of IT st not {A,B,C} is_collinear & {A,B,C} on P & {A,B,C} on Q holds
P = Q ) & ( for L being LINE of IT
for P being PLANE of IT st ex A, B being POINT of IT st
( A <> B & {A,B} on L & {A,B} on P ) holds
L on P ) & ( for A being POINT of IT
for P, Q being PLANE of IT st A on P & A on Q holds
ex B being POINT of IT st
( A <> B & B on P & B on Q ) ) & not for A, B, C, D being POINT of IT holds {A,B,C,D} is_coplanar & ( for A being POINT of IT
for L being LINE of IT
for P being PLANE of IT st A on L & L on P holds
A on P ) );
end;

:: deftheorem Def8 defines IncSpace-like INCSP_1:def 8 :
for IT being IncStruct holds
( IT is IncSpace-like iff ( ( for L being LINE of IT ex A, B being POINT of IT st
( A <> B & {A,B} on L ) ) & ( for A, B being POINT of IT ex L being LINE of IT st {A,B} on L ) & ( for A, B being POINT of IT
for K, L being LINE of IT st A <> B & {A,B} on K & {A,B} on L holds
K = L ) & ( for P being PLANE of IT ex A being POINT of IT st A on P ) & ( for A, B, C being POINT of IT ex P being PLANE of IT st {A,B,C} on P ) & ( for A, B, C being POINT of IT
for P, Q being PLANE of IT st not {A,B,C} is_collinear & {A,B,C} on P & {A,B,C} on Q holds
P = Q ) & ( for L being LINE of IT
for P being PLANE of IT st ex A, B being POINT of IT st
( A <> B & {A,B} on L & {A,B} on P ) holds
L on P ) & ( for A being POINT of IT
for P, Q being PLANE of IT st A on P & A on Q holds
ex B being POINT of IT st
( A <> B & B on P & B on Q ) ) & not for A, B, C, D being POINT of IT holds {A,B,C,D} is_coplanar & ( for A being POINT of IT
for L being LINE of IT
for P being PLANE of IT st A on L & L on P holds
A on P ) ) );

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 S being IncSpace
for L being LINE of S
for P being PLANE of S
for F being Subset of the Points of S st F on L & L on P holds
F on P
proof end;

theorem Th36: :: INCSP_1:36
for S being IncSpace
for A, B being POINT of S holds {A,A,B} is_collinear
proof end;

theorem Th37: :: INCSP_1:37
for S being IncSpace
for A, B, C being POINT of S holds {A,A,B,C} is_coplanar
proof end;

theorem Th38: :: INCSP_1:38
for S being IncSpace
for A, B, C, D being POINT of S st {A,B,C} is_collinear holds
{A,B,C,D} is_coplanar
proof end;

theorem Th39: :: INCSP_1:39
for S being IncSpace
for A, B, C being POINT of S
for L being LINE of S st A <> B & {A,B} on L & not C on L holds
not {A,B,C} is_collinear
proof end;

theorem Th40: :: INCSP_1:40
for S being IncSpace
for A, B, C, D being POINT of S
for P being PLANE of S st not {A,B,C} is_collinear & {A,B,C} on P & not D on P holds
not {A,B,C,D} is_coplanar
proof end;

theorem Th41: :: INCSP_1:41
for S being IncSpace
for K, L being LINE of S st ( for P being PLANE of S holds
( not K on P or not L on P ) ) holds
K <> L
proof end;

Lemma172: for S being IncSpace
for A being POINT of S
for L being LINE of S ex B being POINT of S st
( A <> B & B on L )
proof end;

theorem Th42: :: INCSP_1:42
for S being IncSpace
for L, L1, L2 being LINE of S st ( for P being PLANE of S holds
( not L on P or not L1 on P or not L2 on P ) ) & ex A being POINT of S st
( A on L & A on L1 & A on L2 ) holds
L <> L1
proof end;

theorem Th43: :: INCSP_1:43
for S being IncSpace
for L1, L2, L being LINE of S
for P being PLANE of S st L1 on P & L2 on P & not L on P & L1 <> L2 holds
for Q being PLANE of S holds
( not L on Q or not L1 on Q or not L2 on Q )
proof end;

theorem Th44: :: INCSP_1:44
for S being IncSpace
for A being POINT of S
for L being LINE of S ex P being PLANE of S st
( A on P & L on P )
proof end;

theorem Th45: :: INCSP_1:45
for S being IncSpace
for K, L being LINE of S st ex A being POINT of S st
( A on K & A on L ) holds
ex P being PLANE of S st
( K on P & L on P )
proof end;

theorem Th46: :: INCSP_1:46
for S being IncSpace
for A, B being POINT of S st A <> B holds
ex L being LINE of S st
for K being LINE of S holds
( {A,B} on K iff K = L )
proof end;

theorem Th47: :: INCSP_1:47
for S being IncSpace
for A, B, C being POINT of S st not {A,B,C} is_collinear holds
ex P being PLANE of S st
for Q being PLANE of S holds
( {A,B,C} on Q iff P = Q )
proof end;

theorem Th48: :: INCSP_1:48
for S being IncSpace
for A being POINT of S
for L being LINE of S st not A on L holds
ex P being PLANE of S st
for Q being PLANE of S holds
( ( A on Q & L on Q ) iff P = Q )
proof end;

theorem Th49: :: INCSP_1:49
for S being IncSpace
for K, L being LINE of S st K <> L & ex A being POINT of S st
( A on K & A on L ) holds
ex P being PLANE of S st
for Q being PLANE of S holds
( ( K on Q & L on Q ) iff P = Q )
proof end;

definition
let S be IncSpace;
let A be POINT of S;
let B be POINT of S;
assume E23: A <> B ;
func Line c2,c3 -> LINE of a1 means :Def9: :: INCSP_1:def 9
{A,B} on it;
correctness
existence
ex b1 being LINE of S st {A,B} on b1
;
uniqueness
for b1, b2 being LINE of S st {A,B} on b1 & {A,B} on b2 holds
b1 = b2
;
by Th17, Def5;
end;

:: deftheorem Def9 defines Line INCSP_1:def 9 :
for S being IncSpace
for A, B being POINT of S st A <> B holds
for b4 being LINE of S holds
( b4 = Line A,B iff {A,B} on b4 );

definition
let S be IncSpace;
let A be POINT of S;
let B be POINT of S;
let C be POINT of S;
assume E23: not {A,B,C} is_collinear ;
func Plane c2,c3,c4 -> PLANE of a1 means :Def10: :: INCSP_1:def 10
{A,B,C} on it;
correctness
existence
ex b1 being PLANE of S st {A,B,C} on b1
;
uniqueness
for b1, b2 being PLANE of S st {A,B,C} on b1 & {A,B,C} on b2 holds
b1 = b2
;
by Th18, Def5;
end;

:: deftheorem Def10 defines Plane INCSP_1:def 10 :
for S being IncSpace
for A, B, C being POINT of S st not {A,B,C} is_collinear holds
for b5 being PLANE of S holds
( b5 = Plane A,B,C iff {A,B,C} on b5 );

definition
let S be IncSpace;
let A be POINT of S;
let L be LINE of S;
assume E23: not A on L ;
func Plane c2,c3 -> PLANE of a1 means :Def11: :: INCSP_1:def 11
( A on it & L on it );
existence
ex b1 being PLANE of S st
( A on b1 & L on b1 )
by Th14;
uniqueness
for b1, b2 being PLANE of S st A on b1 & L on b1 & A on b2 & L on b2 holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines Plane INCSP_1:def 11 :
for S being IncSpace
for A being POINT of S
for L being LINE of S st not A on L holds
for b4 being PLANE of S holds
( b4 = Plane A,L iff ( A on b4 & L on b4 ) );

definition
let S be IncSpace;
let K be LINE of S;
let L be LINE of S;
assume that
E23: K <> L and
E24: ex A being POINT of S st
( A on K & A on L ) ;
func Plane c2,c3 -> PLANE of a1 means :Def12: :: INCSP_1:def 12
( K on it & L on it );
existence
ex b1 being PLANE of S st
( K on b1 & L on b1 )
by Th21, Th15;
uniqueness
for b1, b2 being PLANE of S st K on b1 & L on b1 & K on b2 & L on b2 holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines Plane INCSP_1:def 12 :
for S being IncSpace
for K, L being LINE of S st K <> L & ex A being POINT of S st
( A on K & A on L ) holds
for b4 being PLANE of S holds
( b4 = Plane K,L iff ( K on b4 & L 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 S being IncSpace
for A, B being POINT of S st A <> B holds
Line A,B = Line B,A
proof end;

theorem Th58: :: INCSP_1:58
for S being IncSpace
for A, B, C being POINT of S st not {A,B,C} is_collinear holds
Plane A,B,C = Plane A,C,B
proof end;

theorem Th59: :: INCSP_1:59
for S being IncSpace
for A, B, C being POINT of S st not {A,B,C} is_collinear holds
Plane A,B,C = Plane B,A,C
proof end;

theorem Th60: :: INCSP_1:60
for S being IncSpace
for A, B, C being POINT of S st not {A,B,C} is_collinear holds
Plane A,B,C = Plane B,C,A
proof end;

theorem Th61: :: INCSP_1:61
for S being IncSpace
for A, B, C being POINT of S st not {A,B,C} is_collinear holds
Plane A,B,C = Plane C,A,B
proof end;

theorem Th62: :: INCSP_1:62
for S being IncSpace
for A, B, C being POINT of S st not {A,B,C} is_collinear holds
Plane A,B,C = Plane C,B,A
proof end;

theorem Th63: :: INCSP_1:63
canceled;

theorem Th64: :: INCSP_1:64
for S being IncSpace
for K, L being LINE of S st K <> L & ex A being POINT of S st
( A on K & A on L ) holds
Plane K,L = Plane L,K
proof end;

theorem Th65: :: INCSP_1:65
for S being IncSpace
for A, B, C being POINT of S st A <> B & C on Line A,B holds
{A,B,C} is_collinear
proof end;

theorem Th66: :: INCSP_1:66
for S being IncSpace
for A, B, C being POINT of S st A <> B & A <> C & {A,B,C} is_collinear holds
Line A,B = Line A,C
proof end;

theorem Th67: :: INCSP_1:67
for S being IncSpace
for A, B, C being POINT of S st not {A,B,C} is_collinear holds
Plane A,B,C = Plane C,(Line A,B)
proof end;

theorem Th68: :: INCSP_1:68
for S being IncSpace
for A, B, C, D being POINT of S st not {A,B,C} is_collinear & D on Plane A,B,C holds
{A,B,C,D} is_coplanar
proof end;

theorem Th69: :: INCSP_1:69
for S being IncSpace
for C, A, B being POINT of S
for L being LINE of S st not C on L & {A,B} on L & A <> B holds
Plane C,L = Plane A,B,C
proof end;

theorem Th70: :: INCSP_1:70
for S being IncSpace
for A, B, C being POINT of S st not {A,B,C} is_collinear holds
Plane A,B,C = Plane (Line A,B),(Line A,C)
proof end;

Lemma186: for S being IncSpace
for P being PLANE of S ex A, B, C, D being POINT of S st
( A on P & not {A,B,C,D} is_coplanar )
proof end;

theorem Th71: :: INCSP_1:71
for S being IncSpace
for P being PLANE of S ex A, B, C being POINT of S st
( {A,B,C} on P & not {A,B,C} is_collinear )
proof end;

theorem Th72: :: INCSP_1:72
for S being IncSpace
for P being PLANE of S ex A, B, C, D being POINT of S st
( A on P & not {A,B,C,D} is_coplanar ) by ;

theorem Th73: :: INCSP_1:73
for S being IncSpace
for A being POINT of S
for L being LINE of S ex B being POINT of S st
( A <> B & B on L ) by Th13;

theorem Th74: :: INCSP_1:74
for S being IncSpace
for A, B being POINT of S
for P being PLANE of S st A <> B holds
ex C being POINT of S st
( C on P & not {A,B,C} is_collinear )
proof end;

theorem Th75: :: INCSP_1:75
for S being IncSpace
for A, B, C being POINT of S st not {A,B,C} is_collinear holds
ex D being POINT of S st not {A,B,C,D} is_coplanar
proof end;

theorem Th76: :: INCSP_1:76
for S being IncSpace
for A being POINT of S
for P being PLANE of S ex B, C being POINT of S st
( {B,C} on P & not {A,B,C} is_collinear )
proof end;

theorem Th77: :: INCSP_1:77
for S being IncSpace
for A, B being POINT of S st A <> B holds
ex C, D being POINT of S st not {A,B,C,D} is_coplanar
proof end;

theorem Th78: :: INCSP_1:78
for S being IncSpace
for A being POINT of S holds
not for B, C, D being POINT of S holds {A,B,C,D} is_coplanar
proof end;

theorem Th79: :: INCSP_1:79
for S being IncSpace
for A being POINT of S
for P being PLANE of S ex L being LINE of S st
( not A on L & L on P )
proof end;

theorem Th80: :: INCSP_1:80
for S being IncSpace
for A being POINT of S
for P being PLANE of S st A on P holds
ex L, L1, L2 being LINE of S st
( L1 <> L2 & L1 on P & L2 on P & not L on P & A on L & A on L1 & A on L2 )
proof end;

theorem Th81: :: INCSP_1:81
for S being IncSpace
for A being POINT of S ex L, L1, L2 being LINE of S st
( A on L & A on L1 & A on L2 & ( for P being PLANE of S holds
( not L on P or not L1 on P or not L2 on P ) ) )
proof end;

theorem Th82: :: INCSP_1:82
for S being IncSpace
for A being POINT of S
for L being LINE of S ex P being PLANE of S st
( A on P & not L on P )
proof end;

theorem Th83: :: INCSP_1:83
for S being IncSpace
for L being LINE of S
for P being PLANE of S ex A being POINT of S st
( A on P & not A on L )
proof end;

theorem Th84: :: INCSP_1:84
for S being IncSpace
for L being LINE of S ex K being LINE of S st
for P being PLANE of S holds
( not L on P or not K on P )
proof end;

theorem Th85: :: INCSP_1:85
for S being IncSpace
for L being LINE of S ex P, Q being PLANE of S st
( P <> Q & L on P & L on Q )
proof end;

theorem Th86: :: INCSP_1:86
canceled;

theorem Th87: :: INCSP_1:87
for S being IncSpace
for A, B being POINT of S
for L being LINE of S
for P being PLANE of S st not L on P & {A,B} on L & {A,B} on P holds
A = B by Def5;

theorem Th88: :: INCSP_1:88
for S being IncSpace
for P, Q being PLANE of S holds
( not P <> Q or for A being POINT of S holds
( not A on P or not A on Q ) or ex L being LINE of S st
for B being POINT of S holds
( ( B on P & B on Q ) iff B on L ) )
proof end;