### Incidence Projective Spaces

by
Wojciech Leonczuk, and
Krzysztof Prazmowski

Copyright (c) 1990 Association of Mizar Users

MML identifier: INCPROJ
[ MML identifier index ]

```environ

vocabulary COLLSP, PRE_TOPC, INCSP_1, RELAT_1, ANPROJ_2, ANALOAF, VECTSP_1,
AFF_2, INCPROJ;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELSET_1, STRUCT_0, PRE_TOPC,
INCSP_1, COLLSP, ANPROJ_2;
constructors INCSP_1, ANPROJ_2, DOMAIN_1, XBOOLE_0;
clusters ANPROJ_2, RELSET_1, STRUCT_0, SUBSET_1, ZFMISC_1, XBOOLE_0;
requirements SUBSET, BOOLE;
theorems COLLSP, ANPROJ_2, TARSKI, RELAT_1, INCSP_1;
schemes RELSET_1;

begin

reserve CPS for proper CollSp,
B for Element of bool the carrier of CPS,
p for Point of CPS,
x, y, z, Y for set;

definition let CPS;
redefine mode LINE of CPS -> Element of bool the carrier of CPS;
coherence
proof
let x be LINE of CPS;
x c= the carrier of CPS
proof
consider a,b being Point of CPS such that
A1: a <> b & x = Line(a,b) by COLLSP:def 7;
now
let z;
assume z in x;
then z in {p: a,b,p is_collinear} by A1,COLLSP:def 5;
then ex p being Point of CPS st z = p & a, b, p is_collinear;
hence z in the carrier of CPS;
end;
hence thesis by TARSKI:def 3;
end;
hence x is Element of bool the carrier of CPS;
end;
end;

definition let CPS;
func ProjectiveLines(CPS) -> set equals :Def1:
{B : B is LINE of CPS};
coherence;
end;

definition let CPS;
cluster ProjectiveLines(CPS) -> non empty;
coherence
proof
A1: ProjectiveLines(CPS) = {B : B is LINE of CPS} by Def1;
consider x being LINE of CPS;
x in ProjectiveLines(CPS) by A1;
hence thesis;
end;
end;

canceled;

theorem Th2:
x is LINE of CPS iff x is Element of ProjectiveLines(CPS)
proof
hereby
assume x is LINE of CPS;
then x in {B : B is LINE of CPS};
hence x is Element of ProjectiveLines(CPS) by Def1;
end;
assume x is Element of ProjectiveLines(CPS);
then x in ProjectiveLines(CPS);
then x in {B : B is LINE of CPS} by Def1;
then ex B st x=B & B is LINE of CPS;
hence x is LINE of CPS;
end;

definition let CPS;
func Proj_Inc(CPS) -> Relation of the carrier of CPS, ProjectiveLines(CPS)
means :Def2:
for x,y holds [x,y] in it iff x in the carrier of CPS &
y in ProjectiveLines(CPS) & ex Y st y=Y & x in Y;
existence
proof
defpred P[set,set] means ex Y st \$2=Y & \$1 in Y;
ex IT being Relation of the carrier of CPS, ProjectiveLines(CPS) st
for x,y holds [x,y] in IT iff x in the carrier of CPS &
y in ProjectiveLines(CPS) & P[x,y] from Rel_On_Set_Ex;
hence thesis;
end;
uniqueness
proof
let R1,R2 be Relation of the carrier of CPS,ProjectiveLines(CPS) such that
A1: for x,y holds [x,y] in R1 iff x in the carrier of CPS &
y in ProjectiveLines(CPS) & ex Y st y=Y & x in Y and
A2: for x,y holds [x,y] in R2 iff x in the carrier of CPS &
y in ProjectiveLines(CPS) & ex Y st y=Y & x in Y;
now let x,y be set;
A3:now
assume [x,y] in R1;
then x in the carrier of CPS & y in ProjectiveLines(CPS) &
ex Y st y=Y & x in Y by A1;
hence [x,y] in R2 by A2;
end;
now assume [x,y] in R2;
then x in the carrier of CPS & y in ProjectiveLines(CPS)
& ex Y st y=Y & x in Y by A2;
hence [x,y] in R1 by A1;
end;
hence [x,y] in R1 iff [x,y] in R2 by A3;
end;
hence R1 = R2 by RELAT_1:def 2;
end;
end;

definition let CPS;
func IncProjSp_of(CPS) -> IncProjStr equals :Def3:
IncProjStr (# the carrier of CPS, ProjectiveLines(CPS), Proj_Inc(CPS) #);
coherence;
end;

definition let CPS;
cluster IncProjSp_of(CPS) -> strict;
coherence
proof
IncProjSp_of(CPS) =
IncProjStr (# the carrier of CPS, ProjectiveLines(CPS), Proj_Inc(CPS) #)
by Def3; hence thesis;
end;
end;

canceled;

theorem Th4:
the Points of IncProjSp_of(CPS) = the carrier of CPS &
the Lines of IncProjSp_of(CPS) = ProjectiveLines(CPS) &
the Inc of IncProjSp_of(CPS) = Proj_Inc(CPS)
proof
IncProjSp_of(CPS) = IncProjStr
(# the carrier of CPS, ProjectiveLines(CPS), Proj_Inc(CPS) #) by Def3;
hence thesis;
end;

theorem
x is Point of CPS iff x is POINT of IncProjSp_of(CPS) by Th4;

theorem Th6:
x is LINE of CPS iff x is LINE of IncProjSp_of(CPS)
proof
hereby
assume x is LINE of CPS;
then x is Element of ProjectiveLines(CPS) by Th2;
hence x is LINE of IncProjSp_of(CPS) by Th4;
end;
assume x is LINE of IncProjSp_of(CPS);
then x is Element of ProjectiveLines(CPS) by Th4;
hence x is LINE of CPS by Th2;
end;

reserve a,b,c,p,q,s for POINT of IncProjSp_of(CPS),
P,Q,S for LINE of IncProjSp_of(CPS),
a',b',c',p',q',r' for Point of CPS,
P' for LINE of CPS;

canceled;

theorem Th8:
s on S iff [s,S] in Proj_Inc(CPS)
proof
the Inc of IncProjSp_of(CPS) = Proj_Inc(CPS) by Th4;
hence thesis by INCSP_1:def 1;
end;

theorem Th9:
p = p' & P = P' implies (p on P iff p' in P')
proof
assume
A1: p = p' & P = P';
hereby
assume p on P;
then [p',P'] in Proj_Inc(CPS) by A1,Th8;
then ex Y st P'= Y & p' in Y by Def2;
hence p' in P';
end;
assume
A2: p' in P';
reconsider P''= P' as LINE of IncProjSp_of(CPS) by Th6;
reconsider P'''= P'' as Element of ProjectiveLines(CPS) by Th4;
[p',P'''] in Proj_Inc(CPS) by A2,Def2;
hence p on P by A1,Th8;
end;

theorem Th10:
ex a', b', c' st a'<>b' & b'<>c' & c'<>a'
proof
consider a'', b'', c'' being Point of CPS such that
A1: not a'',b'',c'' is_collinear by COLLSP:def 6;
a''<>b'' & b''<>c'' & c''<>a'' by A1,COLLSP:7;
hence thesis;
end;

theorem Th11:
ex b' st a'<>b'
proof
consider p',q',r' such that
A1: p'<>q' & q'<>r' & r'<>p' by Th10;
a'<>p' or a'<>q' by A1;
hence thesis;
end;

theorem Th12:
p on P & q on P & p on Q & q on Q implies p = q or P = Q
proof
assume that
A1: p on P & q on P & p on Q & q on Q and
A2: p<>q;
reconsider p'= p, q'= q as Point of CPS by Th4;
reconsider P'= P, Q'= Q as LINE of CPS by Th6;
p' in P' & q' in P' & p' in Q' & q' in Q' by A1,Th9;
hence thesis by A2,COLLSP:29;
end;

theorem Th13:
ex P st p on P & q on P
proof
reconsider p'= p,q'= q as Point of CPS by Th4;
A1:
now
assume p'<>q';
then consider P' being LINE of CPS such that
A2: p' in P' & q' in P' by COLLSP:24;
reconsider P = P' as LINE of IncProjSp_of(CPS) by Th6;
p on P & q on P by A2,Th9;
hence thesis;
end;
now assume
A3: p'=q';
consider r' such that A4:
p'<>r' by Th11;
consider P' being LINE of CPS such that
A5: p' in P' & r' in P' by A4,COLLSP:24;
reconsider P = P' as LINE of IncProjSp_of(CPS) by Th6;
p on P & q on P by A3,A5,Th9;
hence thesis;
end;
hence thesis by A1;
end;

theorem Th14:
a = a' & b = b' & c = c'
implies
(a',b',c' is_collinear iff ex P st a on P & b on P & c on P)
proof
assume
A1: a = a' & b = b' & c = c';
hereby assume
A2: a',b',c' is_collinear;
A3:now assume
A4: a'<>b'; set X = Line(a',b');
A5: a' in X & b' in X & c' in X by A2,COLLSP:16,17;
reconsider P'= X as LINE of CPS by A4,COLLSP:def 7;
reconsider P = P' as LINE of IncProjSp_of(CPS) by Th6;
a on P & b on P & c on P by A1,A5,Th9;
hence ex P st a on P & b on P & c on P;
end;
now
assume
A6: a'=b';
ex P st b on P & c on P by Th13;
hence ex P st a on P & b on P & c on P by A1,A6;
end;
hence ex P st a on P & b on P & c on P by A3;
end;
given P such that
A7: a on P & b on P & c on P;
reconsider P'=P as LINE of CPS by Th6;
a' in P' & b' in P' & c' in P' by A1,A7,Th9;
hence a',b',c' is_collinear by COLLSP:25;
end;

theorem Th15:
ex p, P st not p on P
proof
consider p',q',r' such that
A1: not p',q',r' is_collinear by COLLSP:def 6;
set X = Line(p',q');
p' <> q' by A1,COLLSP:7;
then reconsider P'= X as LINE of CPS by COLLSP:def 7;
reconsider P = P' as LINE of IncProjSp_of(CPS) by Th6;
reconsider r = r' as POINT of IncProjSp_of(CPS) by Th4;
not r on P
proof
assume not thesis;
then r' in X by Th9;
end;
hence thesis;
end;

reserve CPS for CollProjectiveSpace;

reserve a,b,c,d,p,q for POINT of IncProjSp_of(CPS),
P,Q,S,M,N for LINE of IncProjSp_of(CPS),
a',b',c',d',p',q' for Point of CPS;

theorem Th16:
ex a, b, c st a<>b & b<>c & c <>a & a on P & b on P & c on P
proof
reconsider P'= P as LINE of CPS by Th6;
consider a'', b'' being Point of CPS such that
A1: a''<>b'' & P' = Line(a'',b'') by COLLSP:def 7;
consider c' such that
A2: a''<>c' & b''<>c' and
A3: a'', b'', c' is_collinear by ANPROJ_2:def 10;
reconsider a=a'', b=b'', c =c' as POINT of IncProjSp_of(CPS) by Th4;
take a,b,c;
thus a<>b & b<>c & c <>a by A1,A2;
a'' in P' & b'' in P' by A1,COLLSP:16;
then A4: a on P & b on P by Th9;
ex Q st a on Q & b on Q & c on Q by A3,Th14;
hence a on P & b on P & c on P by A1,A4,Th12;
end;

theorem Th17:
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
implies
ex q st q on P & q on Q
proof
assume that
A1: a on M & b on M and
A2: c on N & d on N and
A3: p on M & p on N and
A4: a on P & c on P and
A5: b on Q & d on Q and
A6: not p on P & not p on Q and
A7: M<>N;
reconsider a'=a,b'=b,c'=c,d'=d,p'=p as Point of CPS by Th4;
b',p',a' is_collinear & p',d',c' is_collinear by A1,A2,A3,Th14;
then consider q' such that
A8: b',d',q' is_collinear & a',c',q' is_collinear by ANPROJ_2:def 9;
reconsider q = q' as POINT of IncProjSp_of(CPS) by Th4;
A9: a<>c by A1,A2,A3,A4,A6,A7,Th12;
A10: b<>d by A1,A2,A3,A5,A6,A7,Th12;
A11: ex P1 being LINE of IncProjSp_of(CPS) st a on P1 & c on P1 & q on P1
by A8,Th14;
ex P2 being LINE of IncProjSp_of(CPS) st b on P2 & d on P2 & q on P2
by A8,Th14;
then q on P & q on Q by A4,A5,A9,A10,A11,Th12;
hence thesis;
end;

theorem Th18:
(for a',b',c',d' ex p' st a', b', p' is_collinear & c', d', p' is_collinear)
implies
for M, N ex q st q on M & q on N
proof
assume
A1: for a',b',c',d' ex p' st a', b', p' is_collinear & c', d', p' is_collinear;
let M, N;
reconsider M'= M, N'= N as LINE of CPS by Th6;
consider a', b' such that
A2: a'<>b' & M'= Line(a',b') by COLLSP:def 7;
consider c', d' such that
A3: c'<>d' & N'= Line(c',d') by COLLSP:def 7;
consider p' such that
A4: a', b', p' is_collinear & c', d', p' is_collinear by A1;
A5: p' in M' & p' in N' by A2,A3,A4,COLLSP:17;
reconsider q = p' as POINT of IncProjSp_of(CPS) by Th4;
q on M & q on N by A5,Th9;
hence thesis;
end;

theorem Th19:
(ex p, p1, r, r1 being Point of CPS st not ex s being Point of CPS st
(p, p1, s is_collinear & r, r1, s is_collinear))
implies
(ex M, N st not ex q st q on M & q on N)
proof
given p, p1, r, r1 being Point of CPS such that
A1: not ex s being Point of CPS st (p,p1,s is_collinear & r,r1,s is_collinear);
set M''= Line(p,p1), N''= Line(r,r1);
p<>p1 & r<>r1
proof
assume
A2: not thesis;
A3:
now
assume p = p1;
then p,p1,r1 is_collinear & r,r1,r1 is_collinear by COLLSP:7;
end;
now
assume r = r1;
then p,p1,p1 is_collinear & r,r1,p1 is_collinear by COLLSP:7;
end;
end;
then reconsider M'= M'', N'= N'' as LINE of CPS by COLLSP:def 7;
reconsider M = M', N = N' as LINE of IncProjSp_of(CPS) by Th6;
take M, N;
thus not ex q st q on M & q on N
proof
assume not thesis;
then consider q such that
A4: q on M & q on N;
reconsider q'=q as Point of CPS by Th4;
q' in M'' & q' in N'' by A4,Th9;
then p,p1,q' is_collinear & r,r1,q' is_collinear by COLLSP:17;
end;
end;

theorem Th20:
(for p, p1, q, q1, r2 being Point of CPS
ex r, r1 being Point of CPS st
p,q,r is_collinear & p1,q1,r1 is_collinear & r2,r,r1 is_collinear)
implies
(for a, M, N ex b, c, S st a on S & b on S & c on S & b on M & c on N)
proof
assume
A1: for p, p1, q, q1, r2 being Point of CPS ex r, r1 being Point of CPS st
p,q,r is_collinear & p1,q1,r1 is_collinear & r2,r,r1 is_collinear;
let a, M, N;
reconsider a'= a as Point of CPS by Th4;
reconsider M'= M, N'= N as LINE of CPS by Th6;
consider p, q being Point of CPS such that
A2: p<>q & M'= Line(p,q) by COLLSP:def 7;
consider p1, q1 being Point of CPS such that
A3: p1<>q1 & N' = Line(p1,q1) by COLLSP:def 7;
consider b', c' such that
A4: p,q,b' is_collinear & p1,q1,c' is_collinear and
A5: a',b',c' is_collinear by A1;
reconsider b = b', c = c' as POINT of IncProjSp_of(CPS) by Th4;
b' in M' & c' in N' by A2,A3,A4,COLLSP:17;
then A6: b on M & c on N by Th9;
ex S st a on S & b on S & c on S by A5,Th14;
hence thesis by A6;
end;

definition let x, y, z be set;
canceled;

pred x,y,z are_mutually_different means
:Def5:
x <> y & y <> z & z <> x;

let u be set;
pred x, y, z, u are_mutually_different means
:Def6:
x <> y & y <> z & z <> x & u <> x & u <> y & u <> z;
end;

definition
let CS be IncProjStr, a,b be POINT of CS,
M be LINE of CS;
pred a,b on M means
:Def7:
a on M & b on M;

let c be POINT of CS;
pred a,b,c on M means
:Def8:
a on M & b on M & c on M;
end;

theorem Th21:
(for p1, r2, q, r1, q1, p, r being Point of CPS holds
(p1,r2,q is_collinear & r1,q1,q is_collinear &
p1,r1,p is_collinear & r2,q1,p is_collinear &
p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear
implies
(p1,r2,q1 is_collinear or p1,r2,r1 is_collinear or
p1,r1,q1 is_collinear or r2,r1,q1 is_collinear)))
implies
(for p, q, r, s, a, b, c being POINT of IncProjSp_of(CPS)
for L, Q, R, S, A, B, C being LINE of IncProjSp_of(CPS) st
not q on L & not r on L & not p on Q & not s on Q & not p on R &
not r on R &
not q on S & not s on S & a,p,s on L & a,q,r on Q & b,q,s on R &
b,p,r on S &
c,p,q on A & c,r,s on B & a,b on C holds not c on C)
proof
assume
A1: for p1, r2, q, r1, q1, p, r being Element of CPS
holds
(p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear &
r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear &
p,q,r is_collinear implies
(p1,r2,q1 is_collinear or p1,r2,r1 is_collinear or p1,r1,q1 is_collinear
or r2,r1,q1 is_collinear));
let p, q, r, s, a, b, c be POINT of IncProjSp_of(CPS);
let L, Q, R, S, A, B, C be LINE of IncProjSp_of(CPS) such that
A2: not q on L & not r on L & not p on Q & not s on Q & not p on R &
not r on R & not q on S & not s on S and
A3: a,p,s on L & a,q,r on Q & b,q,s on R & b,p,r on S & c,p,q on A &
c,r,s on B & a,b on C;
assume
A4: not thesis;
reconsider p'=p, q'=q, r'=r, s'=s, a'=a, b'=b, c'=c as Point of CPS by Th4;
a on C & b on C by A3,Def7;
then A5: a',b',c' is_collinear by A4,Th14;
A6: a on L & p on L & s on L & a on Q & q on Q & r on Q & b on R & q on R &
s on R & b on S & p on S & r on S &
c on A & p on A & q on A & c on B & r on B & s on B by A3,Def8;
then A7: p',r',b' is_collinear & s',q',b' is_collinear & p',s',a' is_collinear
&
r',q',a' is_collinear & p',q',c' is_collinear & r',s',c' is_collinear
by Th14;
A8:
now
assume p',r',q' is_collinear;
then A9: ex K being LINE of IncProjSp_of(CPS) st p on K & r on K & q on K
by Th14;
hence q on S by A2,A6,Th12;
end;
A10:
now
assume p',r',s' is_collinear;
then A11: ex K being LINE of IncProjSp_of(CPS) st p on K & r on K & s on K
by Th14;
hence s on S by A2,A6,Th12;
end;
A12:
now
assume p',s',q' is_collinear;
then A13: ex K being LINE of IncProjSp_of(CPS) st p on K & s on K & q on K
by Th14;
hence p on R by A2,A6,Th12;
end;
now
assume r',s',q' is_collinear;
then A14: ex K being LINE of IncProjSp_of(CPS) st r on K & s on K & q on K
by Th14;
hence r on R by A2,A6,Th12;
end;
end;

theorem Th22:
(for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Point of CPS st
o<>q1 & p1<>q1 & o<>q2 & p2<>q2 & o<>q3 & p3<>q3 &
not o,p1,p2 is_collinear & not o,p1,p3 is_collinear &
not o,p2,p3 is_collinear &
p1,p2,r3 is_collinear & q1,q2,r3 is_collinear &
p2,p3,r1 is_collinear & q2,q3,r1 is_collinear &
p1,p3,r2 is_collinear & q1,q3,r2 is_collinear &
o,p1,q1 is_collinear & o,p2,q2 is_collinear & o,p3,q3 is_collinear
holds
r1,r2,r3 is_collinear)
implies
(for o,b1,a1,b2,a2,b3,a3,r,s,t being POINT of IncProjSp_of(CPS)
for C1,C2,C3,A1,A2,A3,B1,B2,B3 being LINE of IncProjSp_of(CPS) st
o,b1,a1 on C1 & o,a2,b2 on C2 & o,a3,b3 on C3 &
a3,a2,t on A1 & a3,r,a1 on A2 & a2,s,a1 on A3 &
t,b2,b3 on B1 & b1,r,b3 on B2 & b1,s,b2 on B3 &
C1,C2,C3 are_mutually_different &
o<>a1 & o<>a2 & o<>a3 & o<>b1 & o<>b2 & o<>b3 & a1<>b1 & a2<>b2 & a3<>b3
holds
ex O being LINE of IncProjSp_of(CPS) st r,s,t on O)
proof
assume
A1: for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Element of CPS
st o<>q1 & p1<>q1 & o<>q2 & p2<>q2 & o<>q3 & p3<>q3 & not o,p1,p2 is_collinear
& not o,p1,p3 is_collinear & not o,p2,p3 is_collinear & p1,p2,r3 is_collinear
& q1,q2,r3 is_collinear & p2,p3,r1 is_collinear & q2,q3,r1 is_collinear &
p1,p3,r2 is_collinear & q1,q3,r2 is_collinear & o,p1,q1 is_collinear &
o,p2,q2 is_collinear & o,p3,q3 is_collinear holds r1,r2,r3 is_collinear;

let o,b1,a1,b2,a2,b3,a3,r,s,t be POINT of IncProjSp_of(CPS);
let C1,C2,C3,A1,A2,A3,B1,B2,B3 be LINE of IncProjSp_of(CPS) such that
A2: o,b1,a1 on C1 & o,a2,b2 on C2 & o,a3,b3 on C3 & a3,a2,t on A1 &
a3,r,a1 on A2 &
a2,s,a1 on A3 & t,b2,b3 on B1 & b1,r,b3 on B2 & b1,s,b2 on B3 and
A3: C1,C2,C3 are_mutually_different and
A4: o<>a1 & o<>a2 & o<>a3 & o<>b1 & o<>b2 & o<>b3 & a1<>b1 & a2<>b2 & a3<>b3;

reconsider o'=o,b1'=b1,a1'=a1,b2'=b2,a2'=a2,b3'=b3,a3'=a3,r'=r,s'=s,t'=t
as Element of CPS by Th4;
A5: o on C1 & b1 on C1 & a1 on C1 & o on C2 & b2 on C2 & a2 on C2 & o on C3 &
b3 on C3 & a3 on C3 &
a3 on A1 & a2 on A1 & t on A1 & a3 on A2 & a1 on A2 & r on A2 & a2 on A3 &
a1 on A3 & s on A3 &
b3 on B1 & b2 on B1 & t on B1 & b3 on B2 & b1 on B2 & r on B2 & b2 on B3 &
b1 on B3 & s on B3 by A2,Def8;
then A6: b1',b2',s' is_collinear & a1',a2',s' is_collinear &
b1',b3',r' is_collinear & a1',a3',r' is_collinear &
b2',b3',t' is_collinear & a2',a3',t' is_collinear by Th14;
A7: o',b1',a1' is_collinear & o',b2',a2' is_collinear &
o',b3',a3' is_collinear by A5,Th14;
not o',b1',b2' is_collinear & not o',b1',b3' is_collinear &
not o',b2',b3' is_collinear
proof
assume
A8: not thesis;
A9:
now
assume o',b1',b2' is_collinear;
then consider K being LINE of IncProjSp_of(CPS) such that
A10: o on K & b1 on K & b2 on K by Th14;
K = C1 & K = C2 by A4,A5,A10,Th12;
end;
A11:
now
assume o',b1',b3' is_collinear;
then consider K being LINE of IncProjSp_of(CPS) such that
A12: o on K & b1 on K & b3 on K by Th14;
K = C1 & K = C3 by A4,A5,A12,Th12;
end;
now
assume o',b2',b3' is_collinear;
then consider K being LINE of IncProjSp_of(CPS) such that
A13: o on K & b2 on K & b3 on K by Th14;
K = C2 & K = C3 by A4,A5,A13,Th12;
end;
end;
then t',r',s' is_collinear by A1,A4,A6,A7;
then consider O being LINE of IncProjSp_of(CPS) such that
A14: t on O & r on O & s on O by Th14;
r,s,t on O by A14,Def8;
hence thesis;
end;

theorem Th23:
(for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Point of CPS st
o<>p2 & o<>p3 & p2<>p3 & p1<>p2 & p1<>p3 & o<>q2 & o<>q3 & q2<>q3 &
q1<>q2 & q1<>q3 & not o,p1,q1 is_collinear & o,p1,p2 is_collinear &
o,p1,p3 is_collinear & o,q1,q2 is_collinear & o,q1,q3 is_collinear &
p1,q2,r3 is_collinear & q1,p2,r3 is_collinear & p1,q3,r2 is_collinear &
p3,q1,r2 is_collinear & p2,q3,r1 is_collinear & p3,q2,r1 is_collinear
holds
r1,r2,r3 is_collinear)
implies
(for o,a1,a2,a3,b1,b2,b3,c1,c2,c3 being POINT of IncProjSp_of(CPS)
for A1,A2,A3,B1,B2,B3,C1,C2,C3 being LINE of IncProjSp_of(CPS) st
o,a1,a2,a3 are_mutually_different & o,b1,b2,b3 are_mutually_different &
A3<>B3 & o on A3 & o on B3 & a2,b3,c1 on A1 & a3,b1,c2 on B1 &
a1,b2,c3 on C1 &
a1,b3,c2 on A2 & a3,b2,c1 on B2 & a2,b1,c3 on C2 &
b1,b2,b3 on A3 & a1,a2,a3 on B3 & c1,c2 on C3 holds c3 on C3)
proof
assume
A1: for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Point of CPS st
o<>p2 & o<>p3 & p2<>p3 & p1<>p2 & p1<>p3 & o<>q2 & o<>q3 & q2<>q3 &
q1<>q2 & q1<>q3 & not o,p1,q1 is_collinear & o,p1,p2 is_collinear &
o,p1,p3 is_collinear & o,q1,q2 is_collinear & o,q1,q3 is_collinear &
p1,q2,r3 is_collinear & q1,p2,r3 is_collinear & p1,q3,r2 is_collinear &
p3,q1,r2 is_collinear & p2,q3,r1 is_collinear & p3,q2,r1 is_collinear
holds r1,r2,r3 is_collinear;

let o,a1,a2,a3,b1,b2,b3,c1,c2,c3 be POINT of IncProjSp_of(CPS);
let A1,A2,A3,B1,B2,B3,C1,C2,C3 be LINE of IncProjSp_of(CPS) such that
A2: o,a1,a2,a3 are_mutually_different & o,b1,b2,b3 are_mutually_different and
A3: A3<>B3 and A4: o on A3 & o on B3 and
A5: a2,b3,c1 on A1 & a3,b1,c2 on B1 & a1,b2,c3 on C1 & a1,b3,c2 on A2 &
a3,b2,c1 on B2 &
a2,b1,c3 on C2 & b1,b2,b3 on A3 & a1,a2,a3 on B3 & c1,c2 on C3;
A6: o<>b3 & o<>a3 & a1<>a2 & b1<>b2 by A2,Def6;
A7: a2 on A1 & b3 on A1 & c1 on A1 & a3 on B1 & b1 on B1 & c2 on B1 &
a1 on C1 & b2 on C1 & c3 on C1 &
a1 on A2 & b3 on A2 & c2 on A2 & a3 on B2 & b2 on B2 & c1 on B2 &
a2 on C2 & b1 on C2 & c3 on C2 &
b1 on A3 & b2 on A3 & b3 on A3 & a1 on B3 & a2 on B3 & a3 on B3 &
c1 on C3 & c2 on C3 by A5,Def7,Def8;
reconsider o'= o, a1'= a1, a2'= a2, a3'= a3, b1'= b1, b2'= b2, b3'= b3,
c1'= c1, c2'= c2, c3'= c3 as Point of CPS by Th4;
A8: c1<>c2
proof
assume
A9: not thesis;
A10: not a3 on A3 by A3,A4,A6,A7,Th12;
not b3 on B3 by A3,A4,A6,A7,Th12;
then A11: A1<>A2 by A6,A7,Th12;
A12: B1<>B2 by A6,A7,A10,Th12;
A13: c1 = b3 by A7,A9,A11,Th12;
c1 = a3 by A7,A9,A12,Th12;
end;

A14: o'<>a2' & o'<>a3' & a2'<>a3' & a1'<>a2' & a1'<>a3' & o'<>b2' & o'<>b3' &
b2'<>b3' & b1'<>b2' & b1'<>b3' by A2,Def6;

A15: not o',a1',b1' is_collinear
proof
assume not thesis;
then consider K being LINE of IncProjSp_of(CPS) such that
A16: o on K & a1 on K & b1 on K by Th14;
o<>a1 & o<>b1 by A2,Def6;
then K = A3 & K = B3 by A4,A7,A16,Th12;
end;

o',a1',a2' is_collinear & o',a1',a3' is_collinear &
o',b1',b2' is_collinear & o',b1',b3' is_collinear &
a1',b2',c3' is_collinear & b1',a2',c3' is_collinear &
a1',b3',c2' is_collinear & a3',b1',c2' is_collinear &
a2',b3',c1' is_collinear & a3',b2',c1' is_collinear by A4,A7,Th14;
then c1',c2',c3' is_collinear by A1,A14,A15;
then ex K being LINE of IncProjSp_of(CPS) st c1 on K & c2 on K & c3 on K by
Th14;
hence thesis by A7,A8,Th12;
end;

definition let IT be IncProjStr;
attr IT is partial means
:Def9:
for p, q being POINT of IT, P, Q being LINE of IT st
p on P & q on P & p on Q & q on Q holds
p = q or P = Q;

attr IT is linear means
:Def10:
for p,q being POINT of IT ex P being LINE of IT st p on P & q on P;

attr IT is up-2-dimensional means
:Def11:
ex p being POINT of IT, P being LINE of IT st not p on P;

attr IT is up-3-rank means
:Def12:
for P being LINE of IT ex a, b, c being POINT of IT st
a <> b & b <> c & c <> a & a on P & b on P & c on P;

attr IT is Vebleian means
:Def13:
for a, b, c, d, p, q being POINT of IT
for M, N, P, Q being LINE of IT 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 IT st q on P & q on Q;
end;

definition
let CPS be CollProjectiveSpace;
cluster IncProjSp_of(CPS) -> partial linear up-2-dimensional
up-3-rank Vebleian;
coherence
proof
set PS = IncProjSp_of(CPS);
A1: for p, q being POINT of PS
for P, Q being LINE of PS st p on P & q on P & p on Q & q on Q
holds p = q or P = Q by Th12;

A2: for p, q being POINT of PS
ex P being LINE of PS st p on P & q on P by Th13;

A3: ex p being POINT of PS, P being LINE of PS
st not p on P by Th15;

A4: for P being LINE of PS
ex a,b,c being POINT of PS
st a <> b & b <> c & c <> a & a on P & b on P & c on P by Th16;

for a, b, c, d, p, q being POINT of PS
for M, N, P, Q being LINE of PS 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 PS st q on P & q on Q by Th17;
hence thesis by A1,A2,A3,A4,Def9,Def10,Def11,Def12,Def13;
end;
end;

definition
cluster strict partial linear up-2-dimensional up-3-rank
Vebleian IncProjStr;
existence
proof
consider CPS;
IncProjSp_of(CPS) is strict partial linear up-2-dimensional
up-3-rank Vebleian;
hence thesis;
end;
end;

definition
mode IncProjSp is partial linear up-2-dimensional up-3-rank Vebleian
IncProjStr;
end;

definition
let CPS be CollProjectiveSpace;
cluster IncProjSp_of(CPS) -> partial linear up-2-dimensional up-3-rank
Vebleian;
coherence;
end;

definition let IT be IncProjSp;
attr IT is 2-dimensional means
:Def14:
for M,N being LINE of IT ex q being POINT of IT st q on M & q on N;
antonym IT is up-3-dimensional;
end;

definition
let IT be IncProjSp;
canceled;

attr IT is at_most-3-dimensional means
:Def16:
for a being POINT of IT, M, N being LINE of IT
ex b,c being POINT of IT, S being LINE of IT st
a on S & b on S & c on S & b on M & c on N;
end;

definition let IT be IncProjSp;
attr IT is 3-dimensional means
IT is at_most-3-dimensional up-3-dimensional;
end;

definition let IT be IncProjSp;
attr IT is Fanoian means
:Def18:
for p,q,r,s,a,b,c being POINT of IT
for L,Q,R,S,A,B,C being LINE of IT st not q on L & not r on L &
not p on Q & not s on Q & not p on R & not r on R & not q on S &
not s on S &
a,p,s on L & a,q,r on Q & b,q,s on R & b,p,r on S & c,p,q on A &
c,r,s on B & a,b on C
holds
not c on C;
end;

definition let IT be IncProjSp;
attr IT is Desarguesian means
:Def19:
for o,b1,a1,b2,a2,b3,a3,r,s,t being POINT of IT
for C1,C2,C3,A1,A2,A3,B1,B2,B3 being LINE of IT st
o,b1,a1 on C1 & o,a2,b2 on C2 & o,a3,b3 on C3 &
a3,a2,t on A1 & a3,r,a1 on A2 & a2,s,a1 on A3 &
t,b2,b3 on B1 & b1,r,b3 on B2 & b1,s,b2 on B3 &
C1,C2,C3 are_mutually_different &
o<>a1 & o<>a2 & o<>a3 & o<>b1 & o<>b2 & o<>b3 & a1<>b1 & a2<>b2 & a3<>b3
holds
ex O being LINE of IT st r,s,t on O;
end;

definition let IT be IncProjSp;
attr IT is Pappian means
:Def20:
for o,a1,a2,a3,b1,b2,b3,c1,c2,c3 being POINT of IT
for A1,A2,A3,B1,B2,B3,C1,C2,C3 being LINE of IT st
o,a1,a2,a3 are_mutually_different & o,b1,b2,b3 are_mutually_different &
A3<>B3 & o on A3 & o on B3 & a2,b3,c1 on A1 & a3,b1,c2 on B1 &
a1,b2,c3 on C1 &
a1,b3,c2 on A2 & a3,b2,c1 on B2 & a2,b1,c3 on C2 &
b1,b2,b3 on A3 & a1,a2,a3 on B3 & c1,c2 on C3
holds
c3 on C3;
end;

definition
cluster Desarguesian Fanoian at_most-3-dimensional up-3-dimensional
IncProjSp;
existence
proof
consider CPS being Fanoian Desarguesian
at_most-3-dimensional up-3-dimensional CollProjectiveSpace;
reconsider CPS' = CPS as CollProjectiveSpace;
take X = IncProjSp_of(CPS');
for p1,r2,q,r1,q1,p,r being Point of CPS' holds
(p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear &
r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear &
p,q,r is_collinear
implies
(p1,r2,q1 is_collinear or p1,r2,r1 is_collinear or p1,r1,q1 is_collinear or
r2,r1,q1 is_collinear)) by ANPROJ_2:def 11;
then A1: for p,q,r,s,a,b,c being POINT of X
for L,Q,R,S,A,B,C being LINE of X st
not q on L & not r on L & not p on Q & not s on Q & not p on R &
not r on R & not q on S &
not s on S & a,p,s on L & a,q,r on Q & b,q,s on R & b,p,r on S &
c,p,q on A &
c,r,s on B & a,b on C holds not c on C by Th21;
for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Point of CPS' st
o<>q1 & p1<>q1 & o<>q2 & p2<>q2 & o<>q3 & p3<>q3 &
not o,p1,p2 is_collinear & not o,p1,p3 is_collinear &
not o,p2,p3 is_collinear &
p1,p2,r3 is_collinear & q1,q2,r3 is_collinear &
p2,p3,r1 is_collinear & q2,q3,r1 is_collinear &
p1,p3,r2 is_collinear & q1,q3,r2 is_collinear &
o,p1,q1 is_collinear & o,p2,q2 is_collinear & o,p3,q3 is_collinear
holds r1,r2,r3 is_collinear by ANPROJ_2:def 12;
then A2: for o,b1,a1,b2,a2,b3,a3,r,s,t being POINT of X
for C1,C2,C3,A1,A2,A3,B1,B2,B3 being LINE of X st
o,b1,a1 on C1 & o,a2,b2 on C2 & o,a3,b3 on C3 &
a3,a2,t on A1 & a3,r,a1 on A2 & a2,s,a1 on A3 &
t,b2,b3 on B1 & b1,r,b3 on B2 & b1,s,b2 on B3 &
C1,C2,C3 are_mutually_different &
o<>a1 & o<>a2 & o<>a3 & o<>b1 & o<>b2 & o<>b3 & a1<>b1 & a2<>b2 & a3<>b3
holds
ex O being LINE of X st r,s,t on O by Th22;
ex p,p1,r,r1 being Point of CPS' st
not ex s being Point of CPS' st
(p,p1,s is_collinear & r,r1,s is_collinear) by ANPROJ_2:def 14;
then A3: ex M,N being LINE of X st not ex q being POINT of X st q on M &
q on N by Th19;
for p,p1,q,q1,r2 being Point of CPS'
ex r,r1 being Point of CPS' st p,q,r is_collinear &
p1,q1,r1 is_collinear & r2,r,r1 is_collinear by ANPROJ_2:def 15;
then for a being POINT of X, M,N being LINE of X
ex b,c being POINT of X, S being LINE of X st
a on S & b on S & c on S & b on M & c on N by Th20;
hence thesis by A1,A2,A3,Def14,Def16,Def18,Def19;
end;
end;

definition
cluster Pappian Fanoian at_most-3-dimensional up-3-dimensional
IncProjSp;
existence
proof
consider CPS being Fanoian Pappian at_most-3-dimensional up-3-dimensional
CollProjectiveSpace;
reconsider CPS' = CPS as CollProjectiveSpace;
take X = IncProjSp_of(CPS');
for p1,r2,q,r1,q1,p,r being Point of CPS' holds
(p1,r2,q is_collinear & r1,q1,q is_collinear &
p1,r1,p is_collinear & r2,q1,p is_collinear &
p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear
implies
(p1,r2,q1 is_collinear or p1,r2,r1 is_collinear or
p1,r1,q1 is_collinear or r2,r1,q1 is_collinear)) by ANPROJ_2:def 11;
then A1: for p,q,r,s,a,b,c being POINT of X
for L,Q,R,S,A,B,C being LINE of X st
not q on L & not r on L & not p on Q & not s on Q & not p on R &
not r on R &
not q on S & not s on S & a,p,s on L & a,q,r on Q & b,q,s on R &
b,p,r on S &
c,p,q on A & c,r,s on B & a,b on C holds not c on C by Th21;
for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Point of CPS' st
o<>p2 & o<>p3 & p2<>p3 & p1<>p2 & p1<>p3 & o<>q2 & o<>q3 &
q2<>q3 & q1<>q2 & q1<>q3 & not o,p1,q1 is_collinear &
o,p1,p2 is_collinear & o,p1,p3 is_collinear &
o,q1,q2 is_collinear & o,q1,q3 is_collinear &
p1,q2,r3 is_collinear & q1,p2,r3 is_collinear &
p1,q3,r2 is_collinear & p3,q1,r2 is_collinear &
p2,q3,r1 is_collinear & p3,q2,r1 is_collinear
holds
r1,r2,r3 is_collinear by ANPROJ_2:def 13;
then A2: for o,a1,a2,a3,b1,b2,b3,c1,c2,c3 being POINT of X
for A1,A2,A3,B1,B2,B3,C1,C2,C3 being LINE of X st
o,a1,a2,a3 are_mutually_different & o,b1,b2,b3 are_mutually_different &
A3<>B3 & o on A3 & o on B3 & a2,b3,c1 on A1 & a3,b1,c2 on B1 &
a1,b2,c3 on C1 &
a1,b3,c2 on A2 & a3,b2,c1 on B2 & a2,b1,c3 on C2 & b1,b2,b3 on A3 &
a1,a2,a3 on B3 & c1,c2 on C3 holds c3 on C3 by Th23;
ex p,p1,r,r1 being Point of CPS' st
not ex s being Point of CPS' st
(p,p1,s is_collinear & r,r1,s is_collinear) by ANPROJ_2:def 14;
then A3: ex M,N being LINE of X st not ex q being POINT of X st q on M &
q on N by Th19;

for p,p1,q,q1,r2 being Point of CPS' ex r,r1 being Point of CPS' st
p,q,r is_collinear & p1,q1,r1 is_collinear &
r2,r,r1 is_collinear by ANPROJ_2:def 15;
then for a being POINT of X, M,N being LINE of X
ex b,c being POINT of X, S being LINE of X st
a on S & b on S & c on S & b on M & c on N by Th20;
hence thesis by A1,A2,A3,Def14,Def16,Def18,Def20;
end;
end;

definition
cluster Desarguesian Fanoian 2-dimensional IncProjSp;
existence
proof
consider CPS being Fanoian Desarguesian CollProjectivePlane;
reconsider CPS' = CPS as CollProjectiveSpace;
take X = IncProjSp_of(CPS');
for p1,r2,q,r1,q1,p,r being Point of CPS' holds
(p1,r2,q is_collinear & r1,q1,q is_collinear &
p1,r1,p is_collinear & r2,q1,p is_collinear &
p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear
implies
(p1,r2,q1 is_collinear or p1,r2,r1 is_collinear or
p1,r1,q1 is_collinear or r2,r1,q1 is_collinear)) by ANPROJ_2:def 11;
then A1: for p,q,r,s,a,b,c being POINT of X
for L,Q,R,S,A,B,C being LINE of X st
not q on L & not r on L & not p on Q & not s on Q & not p on R & not
r on R & not q on S &
not s on S & a,p,s on L & a,q,r on Q & b,q,s on R & b,p,r on S &
c,p,q on A &
c,r,s on B & a,b on C holds not c on C by Th21;
for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Point of CPS' st
o<>q1 & p1<>q1 & o<>q2 & p2<>q2 & o<>q3 & p3<>q3 &
not o,p1,p2 is_collinear & not o,p1,p3 is_collinear &
not o,p2,p3 is_collinear &
p1,p2,r3 is_collinear & q1,q2,r3 is_collinear &
p2,p3,r1 is_collinear & q2,q3,r1 is_collinear &
p1,p3,r2 is_collinear & q1,q3,r2 is_collinear &
o,p1,q1 is_collinear & o,p2,q2 is_collinear & o,p3,q3 is_collinear
holds
r1,r2,r3 is_collinear by ANPROJ_2:def 12;
then A2: for o,b1,a1,b2,a2,b3,a3,r,s,t being POINT of X
for C1,C2,C3,A1,A2,A3,B1,B2,B3 being LINE of X st
o,b1,a1 on C1 & o,a2,b2 on C2 & o,a3,b3 on C3 &
a3,a2,t on A1 & a3,r,a1 on A2 & a2,s,a1 on A3 &
t,b2,b3 on B1 & b1,r,b3 on B2 & b1,s,b2 on B3 &
C1,C2,C3 are_mutually_different &
o<>a1 & o<>a2 & o<>a3 & o<>b1 & o<>b2 & o<>b3 & a1<>b1 & a2<>b2 & a3<>b3
holds
ex O being LINE of X st r,s,t on O by Th22;
for a,b,c,d being Point of CPS' ex p being Point of CPS' st
a,b,p is_collinear & c,d,p is_collinear by ANPROJ_2:def 14;
then for M,N being LINE of X ex q being POINT of X st q on M & q on N
by Th18;
hence thesis by A1,A2,Def14,Def18,Def19;
end;
end;

definition
cluster Pappian Fanoian 2-dimensional IncProjSp;
existence
proof
consider CPS being Fanoian Pappian CollProjectivePlane;
reconsider CPS' = CPS as CollProjectiveSpace;
take X = IncProjSp_of(CPS');
for p1,r2,q,r1,q1,p,r being Point of CPS' holds
(p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear &
r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear &
p,q,r is_collinear
implies
(p1,r2,q1 is_collinear or p1,r2,r1 is_collinear or
p1,r1,q1 is_collinear or r2,r1,q1 is_collinear)) by ANPROJ_2:def 11;
then A1: for p,q,r,s,a,b,c being POINT of X
for L,Q,R,S,A,B,C being LINE of X st
not q on L & not r on L & not p on Q & not s on Q & not p on R &
not r on R &
not q on S & not s on S & a,p,s on L & a,q,r on Q & b,q,s on R &
b,p,r on S &
c,p,q on A & c,r,s on B & a,b on C holds not c on C by Th21;
for o,p1,p2,p3,q1,q2,q3,r1,r2,r3 being Point of CPS' st
o<>p2 & o<>p3 & p2<>p3 & p1<>p2 & p1<>p3 & o<>q2 & o<>q3 & q2<>q3 &
q1<>q2 & q1<>q3 & not o,p1,q1 is_collinear & o,p1,p2 is_collinear &
o,p1,p3 is_collinear & o,q1,q2 is_collinear & o,q1,q3 is_collinear &
p1,q2,r3 is_collinear & q1,p2,r3 is_collinear & p1,q3,r2 is_collinear &
p3,q1,r2 is_collinear & p2,q3,r1 is_collinear & p3,q2,r1 is_collinear
holds
r1,r2,r3 is_collinear by ANPROJ_2:def 13;
then A2: for o,a1,a2,a3,b1,b2,b3,c1,c2,c3 being POINT of X
for A1,A2,A3,B1,B2,B3,C1,C2,C3 being LINE of X st
o,a1,a2,a3 are_mutually_different & o,b1,b2,b3 are_mutually_different &
A3<>B3 & o on A3 & o on B3 & a2,b3,c1 on A1 & a3,b1,c2 on B1 &
a1,b2,c3 on C1 &
a1,b3,c2 on A2 & a3,b2,c1 on B2 & a2,b1,c3 on C2 &
b1,b2,b3 on A3 & a1,a2,a3 on B3 & c1,c2 on C3
holds
c3 on C3 by Th23;
for a,b,c,d being Point of CPS' ex p being Point of CPS' st
a,b,p is_collinear & c,d,p is_collinear by ANPROJ_2:def 14;
then for M,N being LINE of X ex q being POINT of X st q on M & q on N by
Th18;
hence thesis by A1,A2,Def14,Def18,Def20;
end;
end;
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