### Projective Planes

by
Michal Muzalewski

Copyright (c) 1994 Association of Mizar Users

MML identifier: PROJPL_1
[ MML identifier index ]

```environ

vocabulary INCSP_1, INCPROJ, ANALOAF, MCART_1, PROJPL_1;
notation ZFMISC_1, SUBSET_1, MCART_1, DOMAIN_1, INCSP_1, INCPROJ;
constructors DOMAIN_1, INCPROJ, MEMBERED, XBOOLE_0;
clusters INCPROJ, MEMBERED, ZFMISC_1, XBOOLE_0;
theorems INCPROJ, PROJRED1, MCART_1;

begin

reserve G for IncProjStr;

::
::    1. PROJECTIVE SPACES
::

reserve a,a1,a2,b,b1,b2,c,d,p,q,r for POINT of G;
reserve A,B,C,D,M,N,P,Q,R for LINE of G;

definition let G,a,P;
redefine pred a on P;
antonym a|'P;
end;

definition let G,a,b,P;
pred a,b|'P means
:Def1: a|'P & b|'P;
end;

definition let G,a,P,Q;
pred a on P,Q means
:Def2: a on P & a on Q;
end;

definition let G,a,P,Q,R;
pred a on P,Q,R means
:Def3: a on P & a on Q & a on R;
end;

theorem
Th1: (a,b on P implies b,a on P)
& (a,b,c on P implies a,c,b on P & b,a,c on P & b,c,a on P & c,a,b on P &
c,b,a on P)
& (a on P,Q implies a on Q,P)
& (a on P,Q,R implies a on P,R,Q & a on Q,P,R & a on Q,R,P &
a on R,P,Q & a on R,Q,P)
proof
thus a,b on P implies b,a on P
proof
assume a,b on P; then a on P & b on P by INCPROJ:def 7;
hence thesis by INCPROJ:def 7;
end;
thus a,b,c on P implies a,c,b on P & b,a,c on P & b,c,a on P & c,a,b on P
& c,b,a on P
proof
assume a,b,c on P; then a on P & b on P & c on P by INCPROJ:def 8;
hence thesis by INCPROJ:def 8;
end;
thus a on P,Q implies a on Q,P
proof
assume a on P,Q; then a on P & a on Q by Def2;
hence thesis by Def2;
end;
assume a on P,Q,R; then a on P & a on Q & a on R by Def3;
hence thesis by Def3;
end;

definition let IT be IncProjStr;
attr IT is configuration means
:Def4: 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;
end;

theorem
Th2: G is configuration iff for p,q,P,Q st p,q on P & p,q on Q holds
p = q or P = Q
proof
hereby assume
A1: G is configuration;
let p,q,P,Q;
assume p,q on P & p,q on Q;
then p on P & q on P & p on Q & q on Q by INCPROJ:def 7;
hence p = q or P = Q by A1,Def4;end;
hereby assume
A2: for p,q,P,Q st p,q on P & p,q on Q holds p = q or P = Q;
now let p,q,P,Q;
assume p on P & q on P & p on Q & q on Q;
then p,q on P & p,q on Q by INCPROJ:def 7;
hence p = q or P = Q by A2;end;
hence G is configuration by Def4;end;
end;

theorem
Th3: G is configuration iff
for p,q,P,Q st p on P,Q & q on P,Q holds p = q or P = Q
proof
hereby assume
A1: G is configuration;
let p,q,P,Q;
assume p on P,Q & q on P,Q;
then p on P & q on P & p on Q & q on Q by Def2;
hence p = q or P = Q by A1,Def4;end;
hereby assume
A2: for p,q,P,Q st p on P,Q & q on P,Q holds p = q or P = Q;
now let p,q,P,Q;
assume p on P & q on P & p on Q & q on Q;
then p on P,Q & q on P,Q by Def2;
hence p = q or P = Q by A2;end;
hence G is configuration by Def4;end;
end;

theorem
Th4: G is IncProjSp iff
G is configuration
& (for p,q ex P st p,q on P)
& (ex p,P st p|'P)
& (for P ex a,b,c st a,b,c are_mutually_different & a,b,c on P)
& (for a,b,c,d,p,M,N,P,Q st a,b,p on M & c,d,p on N & a,c on P & b,d on Q &
p|'P & p|'Q
& M<>N holds ex q st q on P,Q)
proof
hereby assume
A1: G is IncProjSp;
then for p,q,P,Q st p on P & q on P & p on Q & q on Q holds p = q or P = Q
by INCPROJ:def 9;
hence G is configuration by Def4;
thus for p,q ex P st p,q on P
proof
let p,q;
consider P such that
A2: p on P & q on P by A1,INCPROJ:def 10;
take P;
thus thesis by A2,INCPROJ:def 7;
end;
thus ex p,P st p|'P by A1,INCPROJ:def 11;
thus for P ex a,b,c st a,b,c are_mutually_different & a,b,c on P
proof
let P;
consider a,b,c such that
A3: a<>b & b<>c & c <>a & a on P & b on P & c on P by A1,INCPROJ:def 12;
take a,b,c;
thus a,b,c are_mutually_different by A3,INCPROJ:def 5;
thus a,b,c on P by A3,INCPROJ:def 8;
end;
thus for a,b,c,d,p,M,N,P,Q st a,b,p on M & c,d,p on N & a,c on P &
b,d on Q & p|'P
& p|'Q & M<>N holds ex q st q on P,Q
proof
let a,b,c,d,p,M,N,P,Q such that
A4: a,b,p on M & c,d,p on N and
A5: a,c on P & b,d on Q and
A6: p|'P & p|'Q & M<>N;
A7: a on M & b on M & p on M & c on N & d on N & p on N
by A4,INCPROJ:def 8;
a on P & c on P & b on Q & d on Q by A5,INCPROJ:def 7;
then consider q such that
A8: q on P & q on Q by A1,A6,A7,INCPROJ:def 13;
take q;
thus thesis by A8,Def2;
end;end;
hereby assume that
A9: G is configuration and
A10: for p,q ex P st p,q on P and
A11: ex p,P st p|'P and
A12: for P ex a,b,c st a,b,c are_mutually_different & a,b,c on P and
A13: for a,b,c,d,p,M,N,P,Q st a,b,p on M & c,d,p on N & a,c on P &
b,d on Q & p|'P
& p|'Q & M<>N holds ex q st q on P,Q;
A14: for p,q,P,Q st p on P & q on P & p on Q & q on Q holds
p = q or P = Q by A9,Def4;
A15: now let p,q; consider P such that
A16: p,q on P by A10;
take P;
thus p on P & q on P by A16,INCPROJ:def 7;
end;
consider p,P such that
A17: p|'P by A11;
A18: now let P; consider a,b,c such that
A19: a,b,c are_mutually_different & a,b,c on P by A12;
A20: a<>b & b<>c & c <>a by A19,INCPROJ:def 5;
a on P & b on P & c on P by A19,INCPROJ:def 8;
hence ex a,b,c st a<>b & b<>c & c <>a & a on P & b on P & c on P by A20;
end;
for a,b,c,d,p,q,M,N,P,Q 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 st q on P & q on Q
proof
let a,b,c,d,p,q,M,N,P,Q such that
A21: 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
and
A22: not p on P & not p on Q & M<>N;
A23: a,b,p on M & c,d,p on N by A21,INCPROJ:def 8;
a,c on P & b,d on Q by A21,INCPROJ:def 7;
then consider q1 being POINT of G such that
A24: q1 on P,Q by A13,A22,A23;
take q1;
thus thesis by A24,Def2;
end;
hence G is IncProjSp by A14,A15,A17,A18,INCPROJ:def 9,def 10,def 11,def 12,
def 13;
end;
end;

definition
mode IncProjectivePlane is 2-dimensional IncProjSp;
end;

definition let G,a,b,c;
pred a,b,c is_collinear means
:Def5: ex P st a,b,c on P;
antonym a,b,c is_a_triangle;
end;

theorem
Th5: a,b,c is_collinear iff ex P st a on P & b on P & c on P
proof
(ex P st a,b,c on P) iff ex P st a on P & b on P & c on P
proof
hereby given P such that
A1: a,b,c on P;
take P;
thus a on P & b on P & c on P by A1,INCPROJ:def 8;end;
hereby given P such that
A2: a on P & b on P & c on P;
take P;
thus a,b,c on P by A2,INCPROJ:def 8;end;
end;
hence thesis by Def5;
end;

theorem
a,b,c is_a_triangle iff for P holds a|'P or b|'P or c|'P by Th5;

definition let G,a,b,c,d;
:Def6: a,b,c is_a_triangle
& b,c,d is_a_triangle
& c,d,a is_a_triangle
& d,a,b is_a_triangle;
end;

theorem
Th7: G is IncProjSp implies ex A,B st A<>B
proof assume
A1: G is IncProjSp;
consider a;
consider A,B,C such that
A2: a on A & a on B & a on C & A<>B & B<>C & C<>A by A1,PROJRED1:5;
take A,B;
thus thesis by A2;
end;

theorem
Th8: G is IncProjSp & a on A implies
ex b,c st b,c on A & a,b,c are_mutually_different
proof assume
A1: G is IncProjSp & a on A;
then consider b such that
A2: b on A & b<>a & b<>a by PROJRED1:8;
consider c such that
A3: c on A & c <>a & c <>b by A1,PROJRED1:8;
take b,c;
thus thesis by A2,A3,INCPROJ:def 5,def 7;
end;

theorem
Th9: G is IncProjSp & a on A & A<>B implies
ex b st b on A & b|'B & a<>b
proof assume
A1: G is IncProjSp & a on A & A<>B;
then consider b,c such that
A2: b,c on A & a,b,c are_mutually_different by Th8;
A3: a<>b & a<>c & b<>c by A2,INCPROJ:def 5;
A4: b on A & c on A by A2,INCPROJ:def 7;
now per cases by A1,A3,A4,INCPROJ:def 9;
case b|'B; ::take b;
hence thesis by A3,A4;
case c|'B; ::take c;
hence thesis by A3,A4;
end;
hence thesis;
end;

theorem
Th10: G is configuration & a1,a2 on A & a1<>a2 & b|'A
implies a1,a2,b is_a_triangle
proof assume
A1: G is configuration & a1,a2 on A & a1<>a2 & b|'A & a1,a2,b is_collinear;
then A2: a1 on A & a2 on A by INCPROJ:def 7;
consider P such that
A3: a1 on P & a2 on P & b on P by A1,Th5;
end;

theorem
Th11: a,b,c is_collinear implies
a,c,b is_collinear
& b,a,c is_collinear
& b,c,a is_collinear
& c,a,b is_collinear
& c,b,a is_collinear
proof assume
a,b,c is_collinear;
then consider P such that
A1: a,b,c on P by Def5;
a,c,b on P & b,a,c on P & b,c,a on P & c,a,b on P & c,b,a on P by A1,Th1;
hence thesis by Def5;
end;

theorem
a,b,c is_a_triangle implies
a,c,b is_a_triangle
& b,a,c is_a_triangle
& b,c,a is_a_triangle
& c,a,b is_a_triangle
& c,b,a is_a_triangle by Th11;

theorem
then A1: a,b,c is_a_triangle
& b,c,d is_a_triangle
& c,d,a is_a_triangle
& d,a,b is_a_triangle by Def6;
then A2: a,c,b is_a_triangle
& b,a,c is_a_triangle
& b,c,a is_a_triangle
& c,a,b is_a_triangle
& c,b,a is_a_triangle by Th11;
A3: b,d,c is_a_triangle
& c,b,d is_a_triangle
& c,d,b is_a_triangle
& d,b,c is_a_triangle
& d,c,b is_a_triangle by A1,Th11;
A4: c,a,d is_a_triangle
& d,c,a is_a_triangle
& d,a,c is_a_triangle
& a,c,d is_a_triangle
& a,d,c is_a_triangle by A1,Th11;
A5: d,b,a is_a_triangle
& a,d,b is_a_triangle
& a,b,d is_a_triangle
& b,d,a is_a_triangle
& b,a,d is_a_triangle by A1,Th11;
end;

theorem
Th14: G is configuration & a1,a2 on A & b1,b2 on B & a1,a2|'B & b1,b2|'A
& a1<>a2 & b1<>b2 implies a1,a2,b1,b2 is_a_quadrangle
proof assume
A1: G is configuration & a1,a2 on A & b1,b2 on B & a1,a2|'B & b1,b2|'A
& a1<>a2 & b1<>b2;
then A2: a1|'B & a2|'B & b1|'A & b2|'A by Def1;
then A3: a1,a2,b1 is_a_triangle by A1,Th10;
b1,b2,a2 is_a_triangle by A1,A2,Th10;
then A4: a2,b1,b2 is_a_triangle by Th11;
A5: b1,b2,a1 is_a_triangle by A1,A2,Th10;
a1,a2,b2 is_a_triangle by A1,A2,Th10;
then b2,a1,a2 is_a_triangle by Th11;
hence thesis by A3,A4,A5,Def6;
end;

theorem
Th15: G is IncProjSp implies ex a,b,c,d st a,b,c,d is_a_quadrangle
proof assume
A1: G is IncProjSp;
then consider A,B such that
A2: A<>B by Th7;
consider a,b such that
A3: a on A & a|'B & b on B & b|'A by A1,A2,PROJRED1:3;
consider p such that
A4: p on A & p|'B & a<>p by A1,A3,Th9;
consider q such that
A5: q on B & q|'A & b<>q by A1,A3,Th9;
A6: G is configuration by A1,Th4;
take a,p,b,q;
A7: a,p on A & b,q on B by A3,A4,A5,INCPROJ:def 7;
a,p|'B & b,q|'A by A3,A4,A5,Def1;
hence thesis by A4,A5,A6,A7,Th14;
end;

definition let G be IncProjSp;
mode Quadrangle of G -> Element of
[:the Points of G,the Points of G,the Points of G,the Points of G:] means
existence
proof
consider a,b,c,d being POINT of G such that
reconsider e = [a,b,c,d] as Element of
[:the Points of G,the Points of G,the Points of G,the Points of G:];
A2: e`1 = a & e`2 = b & e`3 = c & e`4 = d by MCART_1:59;
take e;
thus thesis by A1,A2;
end;
end;

definition let G be IncProjSp, a,b be POINT of G;
assume A1: a <> b;
func a*b -> LINE of G means
:Def8: a,b on it;
existence by Th4;
uniqueness
proof
G is configuration by Th4;
hence thesis by A1,Th2;
end;
end;

theorem Th16: for G be IncProjSp, a,b be POINT of G, L be LINE of G
st a <> b holds a on a*b & b on a*b & a*b = b*a &
(a on L & b on L implies L = a*b)
proof let G be IncProjSp, a,b be POINT of G, L be LINE of G;
assume A1: a <> b;
then a,b on a*b by Def8;
then b,a on a*b by Th1;
hence a on a*b & b on a*b & a*b = b*a by A1,Def8,INCPROJ:def 7;
assume a on L & b on L; then a,b on L by INCPROJ:def 7;
hence thesis by A1,Def8;
end;

::
::    2. PROJECTIVE PLANES
::
begin

theorem
Th17: (ex a,b,c st a,b,c is_a_triangle) & (for p,q ex M st p,q on M)
implies ex p,P st p|'P
proof assume that
A1: ex a,b,c st a,b,c is_a_triangle and
A2: for p,q ex M st p,q on M;
consider a,b,c such that
A3:  a,b,c is_a_triangle by A1;
consider P such that
A4: a,b on P by A2;
A5: a on P & b on P by A4,INCPROJ:def 7;
take c,P;
thus thesis by A3,A5,Th5;
end;

theorem
Th18: (ex a,b,c,d st a,b,c,d is_a_quadrangle)
implies (ex a,b,c st a,b,c is_a_triangle)
proof given a,b,c,d such that
take a,b,c;
thus thesis by A1,Def6;
end;

theorem
Th19: a,b,c is_a_triangle & a,b on P & a,c on Q implies P<>Q
proof assume that
A1: a,b,c is_a_triangle and
A2: a,b on P & a,c on Q;
a on P & b on P & a on Q & c on Q by A2,INCPROJ:def 7;
hence thesis by A1,Th5;
end;

theorem
Th20: a,b,c,d is_a_quadrangle & a,b on P & a,c on Q & a,d on R
implies P,Q,R are_mutually_different
proof assume that
A2: a,b on P & a,c on Q & a,d on R;
a,b,c is_a_triangle by A1,Def6;
then A3: P<>Q by A2,Th19;
c,d,a is_a_triangle by A1,Def6;
then a,c,d is_a_triangle by Th11;
then A4: Q<>R by A2,Th19;
d,a,b is_a_triangle by A1,Def6;
then a,d,b is_a_triangle by Th11;
then R<>P by A2,Th19;
hence thesis by A3,A4,INCPROJ:def 5;
end;

theorem
Th21: G is configuration
& a on P,Q,R & P,Q,R are_mutually_different &
a|'A & p on A,P & q on A,Q & r on A,R
implies p,q,r are_mutually_different
proof assume that
A1: G is configuration and
A2: a on P,Q,R & P,Q,R are_mutually_different &
a|'A & p on A,P & q on A,Q & r on A,R;
A3: a on P & a on Q & a on R by A2,Def3;
A4: P<>Q & Q<>R & R<>P by A2,INCPROJ:def 5;
A5: p on A & p on P & q on A & q on Q & r on A & r on R by A2,Def2;
then A6: p<>q by A1,A2,A3,A4,Def4;
A7: q<>r by A1,A2,A3,A4,A5,Def4;
r<>p by A1,A2,A3,A4,A5,Def4;
hence p,q,r are_mutually_different by A6,A7,INCPROJ:def 5;
end;

theorem
Th22: G is configuration
& (for p,q ex M st p,q on M)
& (for P,Q ex a st a on P,Q)
& (ex a,b,c,d st a,b,c,d is_a_quadrangle)
implies for P ex a,b,c st a,b,c are_mutually_different & a,b,c on P
proof assume that
A1: G is configuration and
A2: for p,q ex M st p,q on M and
A3: for P,Q ex a st a on P,Q and
A4: ex a,b,c,d st a,b,c,d is_a_quadrangle;
hereby let P;
consider a,b,c,d such that
A6: a,b,c is_a_triangle by A5,Def6;
thus ex a,b,c st a,b,c are_mutually_different & a,b,c on P
proof
now per cases by A6,Th5;
case A7: a|'P;
consider B such that
A8: a,b on B by A2;
consider C such that
A9: a,c on C by A2;
consider D such that
A10: a,d on D by A2;
a on B & a on C & a on D by A8,A9,A10,INCPROJ:def 7;
then A11: a on B,C,D by Def3;
A12: B,C,D are_mutually_different by A5,A8,A9,A10,Th20;
consider p such that
A13: p on P,B by A3;
consider q such that
A14: q on P,C by A3;
consider r such that
A15: r on P,D by A3;
p on P & q on P & r on P by A13,A14,A15,Def2;
then A16: p,q,r on P by INCPROJ:def 8;
p,q,r are_mutually_different by A1,A7,A11,A12,A13,A14,A15,Th21;
hence thesis by A16;
case A17: b|'P;
consider B such that
A18: b,a on B by A2;
consider C such that
A19: b,c on C by A2;
consider D such that
A20: b,d on D by A2;
b on B & b on C & b on D by A18,A19,A20,INCPROJ:def 7;
then A21: b on B,C,D by Def3;
then A22: B,C,D are_mutually_different by A18,A19,A20,Th20;
consider p such that
A23: p on P,B by A3;
consider q such that
A24: q on P,C by A3;
consider r such that
A25: r on P,D by A3;
p on P & q on P & r on P by A23,A24,A25,Def2;
then A26: p,q,r on P by INCPROJ:def 8;
p,q,r are_mutually_different by A1,A17,A21,A22,A23,A24,A25,Th21;
hence thesis by A26;
case A27: c|'P;
consider B such that
A28: c,a on B by A2;
consider C such that
A29: c,b on C by A2;
consider D such that
A30: c,d on D by A2;
c on B & c on C & c on D by A28,A29,A30,INCPROJ:def 7;
then A31: c on B,C,D by Def3;
then A32: B,C,D are_mutually_different by A28,A29,A30,Th20;
consider p such that
A33: p on P,B by A3;
consider q such that
A34: q on P,C by A3;
consider r such that
A35: r on P,D by A3;
p on P & q on P & r on P by A33,A34,A35,Def2;
then A36: p,q,r on P by INCPROJ:def 8;
p,q,r are_mutually_different
by A1,A27,A31,A32,A33,A34,A35,Th21;
hence thesis by A36;end;
hence thesis;
end;end;
end;

theorem
Th23: G is IncProjectivePlane iff
G is configuration
& (for p,q ex M st p,q on M)
& (for P,Q ex a st a on P,Q)
& (ex a,b,c,d st a,b,c,d is_a_quadrangle)
proof
hereby assume
A1: G is IncProjectivePlane;
hence G is configuration by Th4;
thus for p,q ex M st p,q on M by A1,Th4;
thus for P,Q ex a st a on P,Q
proof
let P,Q;
consider a such that
A2: a on P & a on Q by A1,INCPROJ:def 14;
take a;
thus thesis by A2,Def2;
end;
thus ex a,b,c,d st a,b,c,d is_a_quadrangle by A1,Th15;end;
hereby assume that
A3: G is configuration and
A4: for p,q ex M st p,q on M and
A5: for P,Q ex a st a on P,Q and
A6: ex a,b,c,d st a,b,c,d is_a_quadrangle;
ex a,b,c st a,b,c is_a_triangle by A6,Th18;
then A7: ex p,P st p|'P by A4,Th17;
A8: for P ex a,b,c st a,b,c are_mutually_different & a,b,c on P
by A3,A4,A5,A6,Th22;
for a,b,c,d,p,M,N,P,Q st a,b,p on M & c,d,p on N & a,c on P & b,d on Q &
p|'P
& p|'Q & M<>N holds ex q st q on P,Q by A5;
then reconsider G'=G as IncProjSp by A3,A4,A7,A8,Th4;
for P,Q being LINE of G' ex a being POINT of G' st a on P & a on Q
proof
let P,Q be LINE of G';
consider a being POINT of G' such that
A9: a on P,Q by A5;
take a;
thus thesis by A9,Def2;
end;
hence G is IncProjectivePlane by INCPROJ:def 14;
end;
end;

reserve G for IncProjectivePlane;

reserve a,q for POINT of G;
reserve A,B for LINE of G;

definition let G,A,B;
assume A1: A <> B;
func A*B -> POINT of G means
:Def9: it on A,B;
existence by Th23;
uniqueness
proof
G is configuration by Th4;
hence thesis by A1,Th3;
end;
end;

theorem Th24: A <> B implies A*B on A & A*B on B & A*B = B*A
& (a on A & a on B implies a=A*B)
proof
assume A1: A <> B;
then A*B on A,B by Def9;
then A*B on B,A by Th1;
hence A*B on A & A*B on B & A*B = B*A by A1,Def2,Def9;
assume a on A & a on B; then a on A,B by Def2;hence thesis by A1,Def9;
end;

theorem A<>B & a on A & q|'A & a<>A*B implies (q*a)*B on B & (q*a)*B|'A
proof assume
A1: A<>B & a on A & q|'A & a<>A*B;
A2: G is configuration by Th23;
A3: a|'B by A1,Th24;
then A4: q*a<>B by A1,Th16;
hence (q*a)*B on B by Th24;
set D=q*a; set d=D*B;
assume A5: d on A;
A6: a on D & q on D by A1,Th16;
d on D by A4,Th24;
then a = d by A1,A2,A5,A6,Def4;
hence thesis by A3,A4,Th24;
end;

::
::    3. SOME USEFUL PROPOSITIONS
::

begin

reserve G for IncProjSp;

reserve a,b,c,d for POINT of G;
reserve P for LINE of G;

theorem
Th26: a,b,c is_a_triangle implies a,b,c are_mutually_different
proof assume that
A1: a,b,c is_a_triangle and
A2: not a,b,c are_mutually_different;
now per cases by A2,INCPROJ:def 5;
case A3: a=b;
consider P such that
A4: b on P & c on P by INCPROJ:def 10;
case A5: b=c;
consider P such that
A6: a on P & b on P by INCPROJ:def 10;
case A7: c =a;
consider P such that
A8: b on P & c on P by INCPROJ:def 10;
hence thesis;
end;

theorem
proof assume that
A2: not a,b,c,d are_mutually_different;
now per cases by A2,INCPROJ:def 6;
case a=b;
then not a,b,c are_mutually_different by INCPROJ:def 5;
then not a,b,c is_a_triangle by Th26;
case b=c;
then not a,b,c are_mutually_different by INCPROJ:def 5;
then not a,b,c is_a_triangle by Th26;
case c =a;
then not a,b,c are_mutually_different by INCPROJ:def 5;
then not a,b,c is_a_triangle by Th26;
case d=a;
then not d,a,b are_mutually_different by INCPROJ:def 5;
then not d,a,b is_a_triangle by Th26;
case d=b;
then not d,a,b are_mutually_different by INCPROJ:def 5;
then not d,a,b is_a_triangle by Th26;
case d=c;
then not b,c,d are_mutually_different by INCPROJ:def 5;
then not b,c,d is_a_triangle by Th26;
hence thesis;
end;

reserve G for IncProjectivePlane;

theorem Th28: for a,b,c,d being POINT of G st a*c = b*d
holds a = c or b = d or c = d or a*c = c*d
proof let a,b,c,d be POINT of G; assume
A1: a*c = b*d & not (a = c or b = d or c = d);
then c on a*c & d on a*c & c on c*d & d on c*d by Th16;
hence thesis by A1,INCPROJ:def 9;
end;

theorem for a,b,c,d being POINT of G st a*c = b*d
holds a = c or b = d or c = d or a on c*d
proof let a,b,c,d be POINT of G; assume
A1: a*c = b*d & not (a = c or b = d or c = d);
then a*c = c*d by Th28;
hence thesis by A1,Th16;
end;

theorem
for G being IncProjectivePlane, e,m,m' being POINT of G,
I being LINE of G st m on I & m' on I & m<>m' & e|'I holds
m*e<>m'*e & e*m<>e*m'
proof let G be IncProjectivePlane, e,m,m' be POINT of G, I be LINE of G
such that
A1: m on I & m' on I & m<>m' & e |'I;
set L1=m*e,L2=m'*e;
A2: now assume
A3: m*e=m'*e;
A4: m on L1 & m' on L2 & e on L1 & e on L2 by A1,Th16;
then m on I,L1 & m' on I,L2 by A1,Def2;
then m=I*L1 & m'=I*L2 by A1,A4,Def9;
now assume
A5: e*m=e*m';
m*e=e*m by A1,Th16;
hence thesis by A2;
end;

theorem
for G being IncProjectivePlane, e being POINT of G,
I,L1,L2 being LINE of G st e on L1 & e on L2 & L1<>L2 & e|'I
holds I*L1<>I*L2 & L1*I<>L2*I
proof let G be IncProjectivePlane, e be POINT of G,
I,L1,L2 be LINE of G such that
A1: e on L1 & e on L2 & L1<>L2 & e |'I;
set p1=I*L1,p2=I*L2;
A2: now assume
A3: I*L1=I*L2;
A4: p1 on I & p2 on I & p1 on L1 & p2 on L2 by A1,Th24;
then e,p1 on L1 & e,p2 on L2 by A1,INCPROJ:def 7;
then L1=e*p1 & L2=e*p2 by A1,A4,Def8;
now assume
A5: L1*I=L2*I;
L1*I=I*L1 by A1,Th24;
hence thesis by A2;
end;

theorem
for G being IncProjSp, a,b,q,q1 being POINT of G
st q on a*b & q on a*q1 & q<>a & q1<>a & a<>b holds q1 on a*b
proof let G be IncProjSp, a,b,q,q1 be POINT of G; assume
A1: q on a*b & q on a*q1 & q<>a & q1<>a & a<>b;
then a on a*b & a on a*q1 by Th16;
then a*b = a*q1 by A1,INCPROJ:def 9;
hence thesis by A1,Th16;
end;

theorem
for G being IncProjSp, a,b,c being POINT of G
st c on a*b & a<>c holds b on a*c
proof let G be IncProjSp, a,b,c be POINT of G; assume
A1: c on a*b & a<>c;
now assume
A2: a<>b;
then a on a*b & a on a*c & c on a*c by A1,Th16;
then a*b = a*c by A1,INCPROJ:def 9;
hence thesis by A2,Th16;end;
hence thesis by A1,Th16;
end;

theorem
for G being IncProjectivePlane, q1,q2,r1,r2 being POINT of G,
H being LINE of G st r1<>r2 & r1 on H & r2 on H & q1|'H & q2|'H
holds q1*r1<>q2*r2
proof let G be IncProjectivePlane, q1,q2,r1,r2 be POINT of G,
H be LINE of G such that
A1: r1<>r2 & r1 on H & r2 on H & q1|'H & q2|'H & q1*r1 = q2*r2;
set L1=q1*r1,L2=q2*r2;
A2: q1 on L1 & q2 on L2 & r1 on L1 & r2 on L2 by A1,Th16;
then r1 on L1,H & r2 on L2,H by A1,Def2;
then r1=L1*H & r2=L2*H by A1,A2,Def9;
end;

theorem
Th35: for a,b,c being POINT of G st a on b*c holds a=c or b=c or b on c*a
proof let a,b,c be POINT of G; assume that
A1: a on b*c and
A2: a<>c and
A3: b<>c;
A4: b on b*c & c on b*c by A3,Th16;
then c,a on b*c by A1,INCPROJ:def 7;
hence thesis by A2,A4,Def8;
end;

theorem
for a,b,c being POINT of G st a on b*c holds b=a or b=c or c on b*a
proof let a,b,c be POINT of G; assume
A1: a on b*c & b<>a & b<>c;
then b*c = c*b by Th16;
hence thesis by A1,Th35;
end;

theorem for e,x1,x2,p1,p2 being POINT of G for H,I being LINE of G
st x1 on I & x2 on I & e on H & e |'I & x1<>x2 & p1<>e & p2 <>e &
p1 on e*x1 & p2 on e*x2
holds ex r being POINT of G st r on p1*p2 & r on H & r<>e
proof let e,x1,x2,p1,p2 be POINT of G; let H,I be LINE of G such that
A1: x1 on I & x2 on I & e on H & e|'I & x1<>x2 & p1<>e & p2<>e &
p1 on e*x1 & p2 on e*x2;
set L1=e*x1,L2=e*x2,L=p1*p2; consider r being POINT of G such that
A2: r on L & r on H by INCPROJ:def 14;
A3: e on L1 & e on L2 & x1 on L1 & x2 on L2 by A1,Th16;
then A4: L1 <> L2 by A1,INCPROJ:def 9;
then p1<>p2 by A1,A3,INCPROJ:def 9;
then A5: p1 on L & p2 on L by Th16;
then L<>L1 by A1,A3,A4,INCPROJ:def 9;
then r<>e by A1,A2,A3,A5,INCPROJ:def 9;
hence thesis by A2;
end;

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