:: A Projective Closure and Projective Horizon of an Affine Space
:: by Henryk Oryszczyszyn and Krzysztof Pra\.zmowski
::
:: Copyright (c) 1990-2019 Association of Mizar Users

:: the projective closure of an affine space. We begin with some evident
:: properties of planes in affine planes.
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 being_plane
proof end;

theorem Th2: :: AFPROJ:2
for AS being AffinSpace
for X being Subset of AS st AS is AffinPlane & X is being_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 being_plane & Y is being_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 being_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 being_line & K is being_line & X is being_plane & Y is being_plane holds
X '||' Y
proof end;

theorem :: AFPROJ:6
for AS being AffinSpace
for A, X, Y being Subset of AS st X is being_plane & A '||' X & X '||' Y holds
A '||' Y by ;

:: Next we distinguish two sets, one consisting of the lines, the second
:: consisting of the planes of a given affine space and we consider two
:: equivalence relations defined on each of these sets; theses relations
:: are in fact the relation of parallelity restricted to suitable area.
:: Their equivalence classes are called directions (of lines and planes,
:: respectively); they are intended to be considered as new objects which
:: extend the given affine space to a projective space.
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 being_line } ;
coherence
{ A where A is Subset of AS : A is being_line } is Subset-Family of AS
proof end;
end;

:: deftheorem defines AfLines AFPROJ:def 1 :
for AS being AffinSpace holds AfLines AS = { A where A is Subset of AS : A is being_line } ;

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 being_plane } ;
coherence
{ A where A is Subset of AS : A is being_plane } is Subset-Family of AS
proof end;
end;

:: deftheorem defines AfPlanes AFPROJ:def 2 :
for AS being AffinSpace holds AfPlanes AS = { A where A is Subset of AS : A is being_plane } ;

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 being_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 being_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 being_line & M is being_line & K '||' M ) } ;
coherence
{ [K,M] where K, M is Subset of AS : ( K is being_line & M is being_line & K '||' M ) } is Equivalence_Relation of (AfLines AS)
proof end;
end;

:: deftheorem defines LinesParallelity AFPROJ:def 3 :
for AS being AffinSpace holds LinesParallelity AS = { [K,M] where K, M is Subset of AS : ( K is being_line & M is being_line & K '||' M ) } ;

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 being_plane & Y is being_plane & X '||' Y ) } ;
coherence
{ [X,Y] where X, Y is Subset of AS : ( X is being_plane & Y is being_plane & X '||' Y ) } is Equivalence_Relation of (AfPlanes AS)
proof end;
end;

:: deftheorem defines PlanesParallelity AFPROJ:def 4 :
for AS being AffinSpace holds PlanesParallelity AS = { [X,Y] where X, Y is Subset of AS : ( X is being_plane & Y is being_plane & X '||' Y ) } ;

definition
let AS be AffinSpace;
let X be Subset of AS;
func LDir X -> Subset of (AfLines AS) equals :: AFPROJ:def 5
Class ((),X);
correctness
coherence
Class ((),X) is Subset of (AfLines AS)
;
;
end;

:: deftheorem defines LDir AFPROJ:def 5 :
for AS being AffinSpace
for X being Subset of AS holds LDir X = Class ((),X);

definition
let AS be AffinSpace;
let X be Subset of AS;
func PDir X -> Subset of (AfPlanes AS) equals :: AFPROJ:def 6
Class ((),X);
correctness
coherence
Class ((),X) is Subset of (AfPlanes AS)
;
;
end;

:: deftheorem defines PDir AFPROJ:def 6 :
for AS being AffinSpace
for X being Subset of AS holds PDir X = Class ((),X);

theorem Th9: :: AFPROJ:9
for AS being AffinSpace
for X being Subset of AS st X is being_line holds
for x being set holds
( x in LDir X iff ex Y being Subset of AS st
( x = Y & Y is being_line & X '||' Y ) )
proof end;

theorem Th10: :: AFPROJ:10
for AS being AffinSpace
for X being Subset of AS st X is being_plane holds
for x being set holds
( x in PDir X iff ex Y being Subset of AS st
( x = Y & Y is being_plane & X '||' Y ) )
proof end;

theorem Th11: :: AFPROJ:11
for AS being AffinSpace
for X, Y being Subset of AS st X is being_line & Y is being_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 being_line & Y is being_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 being_plane & Y is being_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 ();
coherence
Class () is non empty set
proof end;
end;

:: deftheorem defines Dir_of_Lines AFPROJ:def 7 :
for AS being AffinSpace holds Dir_of_Lines AS = Class ();

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

:: deftheorem defines Dir_of_Planes AFPROJ:def 8 :
for AS being AffinSpace holds Dir_of_Planes AS = Class ();

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 being_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 being_plane ) )
proof end;

:: The point is to guarantee that the classes of new objects consist of really
:: new objects. Clearly the set of directions of lines and the set of affine
:: points do not intersect. However we cannot claim, in general, that the set
:: of affine lines and the set of directions of planes do not intersect; this
:: is evidently true only in the case of affine planes. Therefore we have to
:: modify (slightly) a construction of the set of lines of the projective
:: closure of affine space, when compared with (naive) intuitions.
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 being_line ) )
proof end;

theorem Th19: :: AFPROJ:19
for AS being AffinSpace
for x being set holds
( x in [:(),{2}:] iff ex X being Subset of AS st
( x = [(PDir X),2] & X is being_plane ) )
proof end;

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

:: deftheorem defines ProjectivePoints AFPROJ:def 9 :
for AS being AffinSpace holds ProjectivePoints AS = the carrier of AS \/ ();

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

:: deftheorem defines ProjectiveLines AFPROJ:def 10 :
for AS being AffinSpace holds ProjectiveLines AS = [:(AfLines AS),{1}:] \/ [:(),{2}:];

definition
let AS be AffinSpace;
func Proj_Inc AS -> Relation of (),() means :Def11: :: AFPROJ:def 11
for x, y being object holds
( [x,y] in it iff ( ex K being Subset of AS st
( K is being_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 being_line & X is being_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) );
existence
ex b1 being Relation of (),() st
for x, y being object holds
( [x,y] in b1 iff ( ex K being Subset of AS st
( K is being_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 being_line & X is being_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) )
proof end;
uniqueness
for b1, b2 being Relation of (),() st ( for x, y being object holds
( [x,y] in b1 iff ( ex K being Subset of AS st
( K is being_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 being_line & X is being_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) ) ) & ( for x, y being object holds
( [x,y] in b2 iff ( ex K being Subset of AS st
( K is being_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 being_line & X is being_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines Proj_Inc AFPROJ:def 11 :
for AS being AffinSpace
for b2 being Relation of (),() holds
( b2 = Proj_Inc AS iff for x, y being object holds
( [x,y] in b2 iff ( ex K being Subset of AS st
( K is being_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 being_line & X is being_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) ) );

definition
let AS be AffinSpace;
func Inc_of_Dir AS -> Relation of (),() means :Def12: :: AFPROJ:def 12
for x, y being object holds
( [x,y] in it iff ex A, X being Subset of AS st
( x = LDir A & y = PDir X & A is being_line & X is being_plane & A '||' X ) );
existence
ex b1 being Relation of (),() st
for x, y being object holds
( [x,y] in b1 iff ex A, X being Subset of AS st
( x = LDir A & y = PDir X & A is being_line & X is being_plane & A '||' X ) )
proof end;
uniqueness
for b1, b2 being Relation of (),() st ( for x, y being object holds
( [x,y] in b1 iff ex A, X being Subset of AS st
( x = LDir A & y = PDir X & A is being_line & X is being_plane & A '||' X ) ) ) & ( for x, y being object holds
( [x,y] in b2 iff ex A, X being Subset of AS st
( x = LDir A & y = PDir X & A is being_line & X is being_plane & A '||' X ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines Inc_of_Dir AFPROJ:def 12 :
for AS being AffinSpace
for b2 being Relation of (),() holds
( b2 = Inc_of_Dir AS iff for x, y being object holds
( [x,y] in b2 iff ex A, X being Subset of AS st
( x = LDir A & y = PDir X & A is being_line & X is being_plane & A '||' X ) ) );

definition
let AS be AffinSpace;
func IncProjSp_of AS -> strict IncProjStr equals :: AFPROJ:def 13
IncProjStr(# (),(),(Proj_Inc AS) #);
correctness
coherence
IncProjStr(# (),(),(Proj_Inc AS) #) is strict IncProjStr
;
;
end;

:: deftheorem defines IncProjSp_of AFPROJ:def 13 :
for AS being AffinSpace holds IncProjSp_of AS = IncProjStr(# (),(),(Proj_Inc AS) #);

definition
let AS be AffinSpace;
func ProjHorizon AS -> strict IncProjStr equals :: AFPROJ:def 14
IncProjStr(# (),(),() #);
correctness
coherence
IncProjStr(# (),(),() #) is strict IncProjStr
;
;
end;

:: deftheorem defines ProjHorizon AFPROJ:def 14 :
for AS being AffinSpace holds ProjHorizon AS = IncProjStr(# (),(),() #);

theorem Th20: :: AFPROJ:20
for AS being AffinSpace
for x being set holds
( x is POINT of () iff ( x is Element of AS or ex X being Subset of AS st
( x = LDir X & X is being_line ) ) )
proof end;

theorem :: AFPROJ:21
for AS being AffinSpace
for x being set holds
( x is Element of the Points of () iff ex X being Subset of AS st
( x = LDir X & X is being_line ) ) by Th14;

theorem Th22: :: AFPROJ:22
for AS being AffinSpace
for x being set st x is Element of the Points of () holds
x is POINT of ()
proof end;

theorem Th23: :: AFPROJ:23
for AS being AffinSpace
for x being set holds
( x is LINE of () iff ex X being Subset of AS st
( ( x = [X,1] & X is being_line ) or ( x = [(PDir X),2] & X is being_plane ) ) )
proof end;

theorem :: AFPROJ:24
for AS being AffinSpace
for x being set holds
( x is Element of the Lines of () iff ex X being Subset of AS st
( x = PDir X & X is being_plane ) ) by Th15;

theorem Th25: :: AFPROJ:25
for AS being AffinSpace
for x being set st x is Element of the Lines of () holds
[x,2] is LINE of ()
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 ()
for A being LINE of () st x = a & [X,1] = A holds
( a on A iff ( X is being_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 ()
for A being LINE of () st x = a & [(PDir X),2] = A holds
not a on A
proof end;

theorem Th28: :: AFPROJ:28
for AS being AffinSpace
for X, Y being Subset of AS
for a being POINT of ()
for A being LINE of () st a = LDir Y & [X,1] = A & Y is being_line & X is being_line holds
( a on A iff Y '||' X )
proof end;

theorem Th29: :: AFPROJ:29
for AS being AffinSpace
for X, Y being Subset of AS
for a being POINT of ()
for A being LINE of () st a = LDir Y & A = [(PDir X),2] & Y is being_line & X is being_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 ()
for A being LINE of () st X is being_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 ()
for A being LINE of () st X is being_line & Y is being_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 X, Y, X9 being Subset of AS
for a being POINT of ()
for A being LINE of () st Y is being_plane & X c= Y & X9 // X & a = LDir X9 & 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 ()
for A being LINE of () st A = [(PDir X),2] & X is being_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 ()
for A being LINE of () st A = [X,1] & X is being_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 a, p being POINT of ()
for A being LINE of () st A = [X,1] & X is being_line & p on A & a on A & a <> p & p is not Element 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 ()
for A being Element of the Lines of () st a = LDir X & A = PDir Y & X is being_line & Y is being_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 ()
for a9 being Element of the Points of ()
for A being Element of the Lines of ()
for A9 being LINE of () st a9 = a & A9 = [A,2] holds
( a on A iff a9 on A9 )
proof end;

theorem Th38: :: AFPROJ:38
for AS being AffinSpace
for a, b being Element of the Points of ()
for A, K being Element of the Lines of () 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 () ex a, b, c being Element of the Points of () 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 () ex A being Element of the Lines of () st
( a on A & b on A )
proof end;

Lm1: for AS being AffinSpace st AS is not AffinPlane holds
ex a being Element of the Points of () ex A being Element of the Lines of () st not a on A

proof end;

Lm2: for AS being AffinSpace
for a, b being POINT of ()
for A, K being LINE of () st a on A & a on K & b on A & b on K & not a = b holds
A = K

proof end;

Lm3: for AS being AffinSpace
for A being LINE of () ex a, b, c being POINT of () st
( a on A & b on A & c on A & a <> b & b <> c & c <> a )

proof end;

Lm4: for AS being AffinSpace
for a, b being POINT of () ex A being LINE of () st
( a on A & b on A )

proof end;

Lm5: for AS being AffinSpace
for A, K being LINE of () st AS is AffinPlane holds
ex a being POINT of () st
( a on A & a on K )

proof end;

Lm6: for AS being AffinSpace holds
not for a being POINT of ()
for A being LINE of () holds a on A

proof end;

theorem Th41: :: AFPROJ:41
for AS being AffinSpace
for x, y being Element of the Points of ()
for X being Element of the Lines of () st x <> y & [x,X] in the Inc of () & [y,X] in the Inc of () holds
ex Y being Element of the Lines of () st X = [Y,2]
proof end;

theorem Th42: :: AFPROJ:42
for AS being AffinSpace
for x being POINT of ()
for X being Element of the Lines of () st [x,[X,2]] in the Inc of () holds
x is Element of the Points of ()
proof end;

Lm7: for AS being AffinSpace
for X, Y, X9, Y9 being Subset of AS
for b, c being POINT of ()
for A, K, M being LINE of () st X is being_line & X9 is being_line & Y is being_plane & X c= Y & X9 c= Y & A = [X,1] & K = [X9,1] & b on A & c on K & b on M & c on M & b <> c & M = [Y9,1] & Y9 is being_line holds
Y9 c= Y

proof end;

Lm8: for AS being AffinSpace
for X, Y, X9, Y9 being Subset of AS
for b, c being POINT of ()
for A, K, M being LINE of () st X is being_line & X9 is being_line & Y is being_plane & X c= Y & X9 c= Y & A = [X,1] & K = [X9,1] & b on A & c on K & b on M & c on M & b <> c & M = [(PDir Y9),2] & Y9 is being_plane holds
( Y9 '||' Y & Y '||' Y9 )

proof end;

theorem Th43: :: AFPROJ:43
for AS being AffinSpace
for X, Y, X9 being Subset of AS
for P, Q being LINE of () st Y is being_plane & X is being_line & X9 is being_line & X c= Y & X9 c= Y & P = [X,1] & Q = [X9,1] holds
ex q being POINT of () st
( q on P & q on Q )
proof end;

Lm9: for AS being AffinSpace
for a, b, c, d, p being POINT of ()
for M, N, P, Q being LINE of () st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N & p is Element of AS holds
ex q being POINT of () st
( q on P & q on Q )

proof end;

Lm10: for AS being AffinSpace
for a, b, c, d, p being POINT of ()
for M, N, P, Q being LINE of () st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N & p is not Element of AS & a is Element of AS holds
ex q being POINT of () st
( q on P & q on Q )

proof end;

Lm11: for AS being AffinSpace
for a, b, c, d, p being POINT of ()
for M, N, P, Q being LINE of () st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N & p is not Element of AS & ( a is Element of AS or b is Element of AS or c is Element of AS or d is Element of AS ) holds
ex q being POINT of () st
( q on P & q on Q )

proof end;

Lm12: for AS being AffinSpace
for a, b, c, d, p being POINT of ()
for M, N, P, Q being LINE of () st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N holds
ex q being POINT of () st
( q on P & q on Q )

proof end;

theorem Th44: :: AFPROJ:44
for AS being AffinSpace
for a, b, c, d, p being Element of the Points of ()
for M, N, P, Q being Element of the Lines of () st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N holds
ex q being Element of the Points of () st
( q on P & q on Q )
proof end;

registration
let AS be AffinSpace;
correctness
coherence ;
proof end;
end;

registration
existence
ex b1 being IncProjSp st
( b1 is strict & b1 is 2-dimensional )
proof end;
end;

registration
let AS be AffinPlane;
correctness
coherence ;
proof end;
end;

theorem :: AFPROJ:45
for AS being AffinSpace st IncProjSp_of AS is 2-dimensional holds
AS is AffinPlane
proof end;

theorem :: AFPROJ:46
for AS being AffinSpace st AS is not AffinPlane holds
ProjHorizon AS is IncProjSp
proof end;

theorem :: AFPROJ:47
for AS being AffinSpace st ProjHorizon AS is IncProjSp holds
not AS is AffinPlane
proof end;

theorem Th48: :: AFPROJ:48
for AS being AffinSpace
for M, N being Subset of AS
for o, a, b, c, a9, b9, c9 being Element of AS st M is being_line & N is being_line & M <> N & o in M & o in N & o <> b & o <> b9 & o <> c9 & b in M & c in M & a9 in N & b9 in N & c9 in N & a,b9 // b,a9 & b,c9 // c,b9 & ( a = b or b = c or a = c ) holds
a,c9 // c,a9
proof end;

theorem :: AFPROJ:49
for AS being AffinSpace st IncProjSp_of AS is Pappian holds
AS is Pappian
proof end;

theorem Th50: :: AFPROJ:50
for AS being AffinSpace
for A, P, C being Subset of AS
for o, a, b, c, a9, b9, c9 being Element of AS st o in A & o in P & o in C & o <> a & o <> b & o <> c & a in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 & ( o = a9 or a = a9 ) holds
b,c // b9,c9
proof end;

theorem :: AFPROJ:51
for AS being AffinSpace st IncProjSp_of AS is Desarguesian holds
AS is Desarguesian
proof end;

theorem :: AFPROJ:52
for AS being AffinSpace st IncProjSp_of AS is Fanoian holds
AS is Fanoian
proof end;