:: AFPROJ semantic presentation

theorem Th1: :: AFPROJ:1

for AS being AffinSpace
for X being Subset of AS st AS is AffinPlane & X = the carrier of AS holds
X is_plane

proof end;

theorem Th2: :: AFPROJ:2

for AS being AffinSpace
for X being Subset of AS st AS is AffinPlane & X is_plane holds
X = the carrier of AS

proof end;

theorem Th3: :: AFPROJ:3

for AS being AffinSpace
for X, Y being Subset of AS st AS is AffinPlane & X is_plane & Y is_plane holds
X = Y

proof end;

theorem :: AFPROJ:4

for AS being AffinSpace
for X being Subset of AS st X = the carrier of AS & X is_plane holds
AS is AffinPlane

proof end;

theorem Th5: :: AFPROJ:5

for AS being AffinSpace
for A, K, X, Y being Subset of AS st not A // K & A '||' X & A '||' Y & K '||' X & K '||' Y & A is_line & K is_line & X is_plane & Y is_plane holds
X '||' Y

proof end;

theorem :: AFPROJ:6

for AS being AffinSpace
for X, A, Y being Subset of AS st X is_plane & A '||' X & X '||' Y holds
A '||' Y by AFF_4:59, AFF_4:60;

definition

let AS be AffinSpace;
func AfLines AS -> Subset-Family of AS equals :: AFPROJ:def 1
{ A where A is Subset of AS : A is_line } ;
coherence

proof end;

end;

:: deftheorem defines AfLines AFPROJ:def 1 :

definition

let AS be AffinSpace;
func AfPlanes AS -> Subset-Family of AS equals :: AFPROJ:def 2
{ A where A is Subset of AS : A is_plane } ;
coherence

proof end;

end;

:: deftheorem defines AfPlanes AFPROJ:def 2 :

theorem :: AFPROJ:7

for AS being AffinSpace
for x being set holds
( x in AfLines AS iff ex X being Subset of AS st
( x = X & X is_line ) ) ;

theorem :: AFPROJ:8

for AS being AffinSpace
for x being set holds
( x in AfPlanes AS iff ex X being Subset of AS st
( x = X & X is_plane ) ) ;

definition

let AS be AffinSpace;
func LinesParallelity AS -> Equivalence_Relation of AfLines AS equals :: AFPROJ:def 3
{ [K,M] where K, M is Subset of AS : ( K is_line & M is_line & K '||' M ) } ;
coherence

proof end;

end;

:: deftheorem defines LinesParallelity AFPROJ:def 3 :

definition

let AS be AffinSpace;
func PlanesParallelity AS -> Equivalence_Relation of AfPlanes AS equals :: AFPROJ:def 4
{ [X,Y] where X, Y is Subset of AS : ( X is_plane & Y is_plane & X '||' Y ) } ;
coherence

proof end;

end;

:: deftheorem defines PlanesParallelity AFPROJ:def 4 :

definition

let AS be AffinSpace;
let X be Subset of AS;
assume X is_line ;
func LDir X -> Subset of (AfLines AS) equals :Def5: :: AFPROJ:def 5
Class (LinesParallelity AS),X;
correctness ;

end;

:: deftheorem Def5 defines LDir AFPROJ:def 5 :

definition

let AS be AffinSpace;
let X be Subset of AS;
assume X is_plane ;
func PDir X -> Subset of (AfPlanes AS) equals :Def6: :: AFPROJ:def 6
Class (PlanesParallelity AS),X;
correctness ;

end;

:: deftheorem Def6 defines PDir AFPROJ:def 6 :

theorem Th9: :: AFPROJ:9

for AS being AffinSpace
for X being Subset of AS st X is_line holds
for x being set holds
( x in LDir X iff ex Y being Subset of AS st
( x = Y & Y is_line & X '||' Y ) )

proof end;

theorem Th10: :: AFPROJ:10

for AS being AffinSpace
for X being Subset of AS st X is_plane holds
for x being set holds
( x in PDir X iff ex Y being Subset of AS st
( x = Y & Y is_plane & X '||' Y ) )

proof end;

theorem Th11: :: AFPROJ:11

for AS being AffinSpace
for X, Y being Subset of AS st X is_line & Y is_line holds
( LDir X = LDir Y iff X // Y )

proof end;

theorem Th12: :: AFPROJ:12

for AS being AffinSpace
for X, Y being Subset of AS st X is_line & Y is_line holds
( LDir X = LDir Y iff X '||' Y )

proof end;

theorem Th13: :: AFPROJ:13

for AS being AffinSpace
for X, Y being Subset of AS st X is_plane & Y is_plane holds
( PDir X = PDir Y iff X '||' Y )

proof end;

definition

let AS be AffinSpace;
func Dir_of_Lines AS -> non empty set equals :: AFPROJ:def 7
Class (LinesParallelity AS);
coherence

proof end;

end;

:: deftheorem defines Dir_of_Lines AFPROJ:def 7 :

definition

let AS be AffinSpace;
func Dir_of_Planes AS -> non empty set equals :: AFPROJ:def 8
Class (PlanesParallelity AS);
coherence

proof end;

end;

:: deftheorem defines Dir_of_Planes AFPROJ:def 8 :

theorem Th14: :: AFPROJ:14

for AS being AffinSpace
for x being set holds
( x in Dir_of_Lines AS iff ex X being Subset of AS st
( x = LDir X & X is_line ) )

proof end;

theorem Th15: :: AFPROJ:15

for AS being AffinSpace
for x being set holds
( x in Dir_of_Planes AS iff ex X being Subset of AS st
( x = PDir X & X is_plane ) )

proof end;

theorem Th16: :: AFPROJ:16

for AS being AffinSpace holds the carrier of AS misses Dir_of_Lines AS

proof end;

theorem :: AFPROJ:17

for AS being AffinSpace st AS is AffinPlane holds
AfLines AS misses Dir_of_Planes AS

proof end;

theorem Th18: :: AFPROJ:18

for AS being AffinSpace
for x being set holds
( x in [:(AfLines AS),{1}:] iff ex X being Subset of AS st
( x = [X,1] & X is_line ) )

proof end;

theorem Th19: :: AFPROJ:19

for AS being AffinSpace
for x being set holds
( x in [:(Dir_of_Planes AS),{2}:] iff ex X being Subset of AS st
( x = [(PDir X),2] & X is_plane ) )

proof end;

definition

let AS be AffinSpace;
func ProjectivePoints AS -> non empty set equals :: AFPROJ:def 9
the carrier of AS \/ (Dir_of_Lines AS);
correctness ;

end;

:: deftheorem defines ProjectivePoints AFPROJ:def 9 :

definition

let AS be AffinSpace;
func ProjectiveLines AS -> non empty set equals :: AFPROJ:def 10
[:(AfLines AS),{1}:] \/ [:(Dir_of_Planes AS),{2}:];
coherence ;

end;

:: deftheorem defines ProjectiveLines AFPROJ:def 10 :

definition

let AS be AffinSpace;
func Proj_Inc AS -> Relation of ProjectivePoints AS, ProjectiveLines AS means :Def11: :: AFPROJ:def 11
for x, y being set holds
( [x,y] in it iff ( ex K being Subset of AS st
( K is_line & y = [K,1] & ( ( x in the carrier of AS & x in K ) or x = LDir K ) ) or ex K, X being Subset of AS st
( K is_line & X is_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def11 defines Proj_Inc AFPROJ:def 11 :

definition

let AS be AffinSpace;
func Inc_of_Dir AS -> Relation of Dir_of_Lines AS, Dir_of_Planes AS means :Def12: :: AFPROJ:def 12
for x, y being set holds
( [x,y] in it iff ex A, X being Subset of AS st
( x = LDir A & y = PDir X & A is_line & X is_plane & A '||' X ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def12 defines Inc_of_Dir AFPROJ:def 12 :

definition

let AS be AffinSpace;
func IncProjSp_of AS -> strict IncProjStr equals :: AFPROJ:def 13
IncProjStr(# (ProjectivePoints AS),(ProjectiveLines AS),(Proj_Inc AS) #);
correctness ;

end;

:: deftheorem defines IncProjSp_of AFPROJ:def 13 :

definition

let AS be AffinSpace;
func ProjHorizon AS -> strict IncProjStr equals :: AFPROJ:def 14
IncProjStr(# (Dir_of_Lines AS),(Dir_of_Planes AS),(Inc_of_Dir AS) #);
correctness ;

end;

:: deftheorem defines ProjHorizon AFPROJ:def 14 :

theorem Th20: :: AFPROJ:20

for AS being AffinSpace
for x being set holds
( x is POINT of (IncProjSp_of AS) iff ( x is Element of AS or ex X being Subset of AS st
( x = LDir X & X is_line ) ) )

proof end;

theorem :: AFPROJ:21

for AS being AffinSpace
for x being set holds
( x is Element of the Points of (ProjHorizon AS) iff ex X being Subset of AS st
( x = LDir X & X is_line ) ) by Th14;

theorem Th22: :: AFPROJ:22

for AS being AffinSpace
for x being set st x is Element of the Points of (ProjHorizon AS) holds
x is POINT of (IncProjSp_of AS)

proof end;

theorem Th23: :: AFPROJ:23

for AS being AffinSpace
for x being set holds
( x is LINE of (IncProjSp_of AS) iff ex X being Subset of AS st
( ( x = [X,1] & X is_line ) or ( x = [(PDir X),2] & X is_plane ) ) )

proof end;

theorem :: AFPROJ:24

for AS being AffinSpace
for x being set holds
( x is Element of the Lines of (ProjHorizon AS) iff ex X being Subset of AS st
( x = PDir X & X is_plane ) ) by Th15;

theorem Th25: :: AFPROJ:25

for AS being AffinSpace
for x being set st x is Element of the Lines of (ProjHorizon AS) holds
[x,2] is LINE of (IncProjSp_of AS)

proof end;

theorem Th26: :: AFPROJ:26

for AS being AffinSpace
for x being Element of AS
for X being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st x = a & [X,1] = A holds
( a on A iff ( X is_line & x in X ) )

proof end;

theorem Th27: :: AFPROJ:27

for AS being AffinSpace
for x being Element of AS
for X being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st x = a & [(PDir X),2] = A & X is_plane holds
not a on A

proof end;

theorem Th28: :: AFPROJ:28

for AS being AffinSpace
for Y, X being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st a = LDir Y & [X,1] = A & Y is_line & X is_line holds
( a on A iff Y '||' X )

proof end;

theorem Th29: :: AFPROJ:29

for AS being AffinSpace
for Y, X being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st a = LDir Y & A = [(PDir X),2] & Y is_line & X is_plane holds
( a on A iff Y '||' X )

proof end;

theorem Th30: :: AFPROJ:30

for AS being AffinSpace
for X being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st X is_line & a = LDir X & A = [X,1] holds
a on A

proof end;

theorem Th31: :: AFPROJ:31

for AS being AffinSpace
for X, Y being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st X is_line & Y is_plane & X c= Y & a = LDir X & A = [(PDir Y),2] holds
a on A

proof end;

theorem Th32: :: AFPROJ:32

for AS being AffinSpace
for Y, X, X' being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st Y is_plane & X c= Y & X' // X & a = LDir X' & A = [(PDir Y),2] holds
a on A

proof end;

theorem :: AFPROJ:33

for AS being AffinSpace
for X being Subset of AS
for a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st A = [(PDir X),2] & X is_plane & a on A holds
not a is Element of AS by Th27;

theorem Th34: :: AFPROJ:34

for AS being AffinSpace
for X being Subset of AS
for p being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st A = [X,1] & X is_line & p on A & p is not Element of AS holds
p = LDir X

proof end;

theorem Th35: :: AFPROJ:35

for AS being AffinSpace
for X being Subset of AS
for p, a being POINT of (IncProjSp_of AS)
for A being LINE of (IncProjSp_of AS) st A = [X,1] & X is_line & p on A & a on A & a <> p & p is not Element of the carrier of AS holds
a is Element of AS

proof end;

theorem Th36: :: AFPROJ:36

for AS being AffinSpace
for X, Y being Subset of AS
for a being Element of the Points of (ProjHorizon AS)
for A being Element of the Lines of (ProjHorizon AS) st a = LDir X & A = PDir Y & X is_line & Y is_plane holds
( a on A iff X '||' Y )

proof end;

theorem Th37: :: AFPROJ:37

for AS being AffinSpace
for a being Element of the Points of (ProjHorizon AS)
for a' being Element of the Points of (IncProjSp_of AS)
for A being Element of the Lines of (ProjHorizon AS)
for A' being LINE of (IncProjSp_of AS) st a' = a & A' = [A,2] holds
( a on A iff a' on A' )

proof end;

theorem Th38: :: AFPROJ:38

for AS being AffinSpace
for a, b being Element of the Points of (ProjHorizon AS)
for A, K being Element of the Lines of (ProjHorizon AS) st a on A & a on K & b on A & b on K & not a = b holds
A = K

proof end;

theorem Th39: :: AFPROJ:39

for AS being AffinSpace
for A being Element of the Lines of (ProjHorizon AS) ex a, b, c being Element of the Points of (ProjHorizon AS) st
( a on A & b on A & c on A & a <> b & b <> c & c <> a )

proof end;

theorem Th40: :: AFPROJ:40

for AS being AffinSpace
for a, b being Element of the Points of (ProjHorizon AS) ex A being Element of the Lines of (ProjHorizon AS) st
( a on A & b on A )

proof end;

Lm1: for AS being AffinSpace st