### Desargues Theorem In Projective 3-Space

by
Eugeniusz Kusak

Copyright (c) 1990 Association of Mizar Users

MML identifier: PROJDES1
[ MML identifier index ]

```environ

vocabulary ANPROJ_2, INCSP_1, AFF_2, PROJDES1;
notation STRUCT_0, COLLSP, ANPROJ_2;
constructors ANPROJ_2, XBOOLE_0;
clusters ANPROJ_2, ZFMISC_1, XBOOLE_0;
theorems ANPROJ_2, COLLSP;

begin

reserve FCPS for up-3-dimensional CollProjectiveSpace;
reserve a,a',b,b',c,c',d,d',o,p,q,r,s,t,u,x,y,z
for Element of FCPS;

theorem
Th1: a,b,c is_collinear implies b,c,a is_collinear & c,a,b is_collinear &
b,a,c is_collinear & a,c,b is_collinear & c,b,a is_collinear
proof assume A1: a,b,c is_collinear; then A2: b,a,c is_collinear by COLLSP:9;
A3: a,c,b is_collinear by A1,COLLSP:9; b,c,a is_collinear by A2,COLLSP:9;
hence thesis by A3,COLLSP:9; end;

canceled 3;

theorem
ex q,r st not p,q,r is_collinear
proof consider q such that A1: p<>q & p<>q & p,p,q is_collinear
by ANPROJ_2:def 10;
ex r st not p,q,r is_collinear by A1,COLLSP:19;
hence thesis; end;

theorem
not a,b,c is_collinear & a,b,b' is_collinear & a<>b' implies
not a,b',c is_collinear
proof assume A1: not a,b,c is_collinear & a,b,b' is_collinear & a<>b'; assume
not thesis; then a,b',b is_collinear & a,b',c is_collinear by A1,Th1;

theorem
not a,b,c is_collinear & a,b,d is_collinear & a,c,d is_collinear
implies a=d
proof assume A1: not a,b,c is_collinear & a,b,d is_collinear &
a,c,d is_collinear; assume A2: not thesis; a,d,a is_collinear &
a,d,b is_collinear & a,d,c is_collinear by A1,Th1,ANPROJ_2:def 7;
hence contradiction by A1,A2,ANPROJ_2:def 8; end;

theorem
Th8: not o,a,d is_collinear & o,d,d' is_collinear & a,d,s is_collinear &
d<>d' & a',d',s is_collinear & o,a,a' is_collinear & o<>a' implies s<>d
proof assume A1: not o,a,d is_collinear & o,d,d' is_collinear &
a,d,s is_collinear & d<>d' & a',d',s is_collinear & o,a,a' is_collinear &
o<>a'; assume not thesis;
then A2: d,d',a' is_collinear by A1,Th1; d,d',o is_collinear by A1,Th1;
then d,o,a' is_collinear by A1,A2,COLLSP:11; then A3: o,a',d is_collinear by
Th1;
o,a',a is_collinear by A1,Th1;

Lm1: not a,b,c is_collinear & a,b,b' is_collinear & a,c,c' is_collinear &
a<>b' implies b'<>c'
proof assume A1: not a,b,c is_collinear & a,b,b' is_collinear &
a,c,c' is_collinear & a<>b'; assume not thesis;
then A2: a,b',c is_collinear by A1,Th1; a,b',b is_collinear by A1,Th1;

definition let FCPS,a,b,c,d;
pred a,b,c,d are_coplanar means
:Def1: ex x being Element of FCPS
st a,b,x is_collinear & c,d,x is_collinear;
end;

canceled;

theorem
Th10: a,b,c is_collinear or b,c,d is_collinear or c,d,a is_collinear or
d,a,b is_collinear implies a,b,c,d are_coplanar
proof assume A1: a,b,c is_collinear or b,c,d is_collinear or c,d,a is_collinear
or d,a,b is_collinear;
A2: now assume A3: a,b,c is_collinear; c,d,c is_collinear by ANPROJ_2:def 7;
hence thesis by A3,Def1; end;
A4: now assume b,c,d is_collinear; then A5: c,d,b is_collinear by Th1;
a,b,b is_collinear by ANPROJ_2:def 7; hence thesis by A5,Def1; end;
A6: now assume A7: c,d,a is_collinear; a,b,a is_collinear by ANPROJ_2:def 7;
hence thesis by A7,Def1; end;
now assume d,a,b is_collinear; then A8: a,b,d is_collinear by Th1;
c,d,d is_collinear by ANPROJ_2:def 7; hence thesis by A8,Def1; end;
hence thesis by A1,A2,A4,A6; end;

Lm2: a,b,c,d are_coplanar implies b,a,c,d are_coplanar
proof assume a,b,c,d are_coplanar; then consider x such that A1:
a,b,x is_collinear & c,d,x is_collinear by Def1; b,a,x is_collinear by A1,Th1
;
hence thesis by A1,Def1; end;

Lm3: a,b,c,d are_coplanar implies b,c,d,a are_coplanar
proof assume a,b,c,d are_coplanar; then consider x such that A1:
a,b,x is_collinear & c,d,x is_collinear by Def1; A2: b,x,a is_collinear
by A1,Th1; x,c,d is_collinear by A1,Th1; then consider y such that
A3: b,c,y is_collinear & a,d,y is_collinear by A2,ANPROJ_2:def 9
; b,c,y is_collinear &
d,a,y is_collinear by A3,Th1; hence thesis by Def1; end;

theorem
Th11: a,b,c,d are_coplanar implies b,c,d,a are_coplanar & c,d,a,b are_coplanar
& d,a,b,c are_coplanar & b,a,c,d are_coplanar & c,b,d,a are_coplanar &
d,c,a,b are_coplanar & a,d,b,c are_coplanar & a,c,d,b are_coplanar &
b,d,a,c are_coplanar & c,a,b,d are_coplanar & d,b,c,a are_coplanar &
c,a,d,b are_coplanar & d,b,a,c are_coplanar & a,c,b,d are_coplanar &
b,d,c,a are_coplanar & a,b,d,c are_coplanar & a,d,c,b are_coplanar &
b,c,a,d are_coplanar & b,a,d,c are_coplanar & c,b,a,d are_coplanar &
c,d,b,a are_coplanar & d,a,c,b are_coplanar & d,c,b,a are_coplanar
proof assume A1: a,b,c,d are_coplanar;
then A2: b,a,c,d are_coplanar by Lm2; then a,c,d,b are_coplanar by Lm3;
then A3: c,d,b,a are_coplanar by Lm3; then A4: d,b,a,c are_coplanar by Lm3;
then b,d,a,c are_coplanar by Lm2; then A5: d,a,c,b are_coplanar by Lm3;
then A6: a,c,b,d are_coplanar by Lm3;
then c,a,b,d are_coplanar by Lm2; then A7: a,b,d,c are_coplanar by Lm3;
then A8: b,d,c,a are_coplanar by Lm3; then A9: d,c,a,b are_coplanar by Lm3;
then c,d,a,b are_coplanar by Lm2; then A10: d,a,b,c are_coplanar by Lm3;
then a,d,b,c are_coplanar by Lm2; then d,b,c,a are_coplanar by Lm3;
then b,c,a,d are_coplanar by Lm3;
hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,Lm2,Lm3; end;

Lm4: not a,b,c is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar
implies a,b,p,q are_coplanar
proof assume A1: not a,b,c is_collinear & a,b,c,p are_coplanar &
a,b,c,q are_coplanar; then consider x such that A2: a,b,x is_collinear &
c,p,x is_collinear by Def1;
consider y such that A3: a,b,y is_collinear & c,q,y is_collinear by A1,Def1;
A4: now assume a=b; then a,b,p is_collinear by ANPROJ_2:def 7
; hence thesis by Th10; end;
now assume A5: a<>b; then A6: a,x,y is_collinear by A2,A3,COLLSP:11;
b,a,x is_collinear & b,a,y is_collinear by A2,A3,Th1; then b,x,y
is_collinear
by A5,COLLSP:11; then A7: x,y,a is_collinear & x,y,b is_collinear by A6,Th1;
A8: now assume x=y; then x,c,p is_collinear & x,c,q is_collinear by A2,A3,Th1
;
then x,p,q is_collinear by A1,A2,COLLSP:11; then p,q,x is_collinear by Th1;
hence
thesis by A2,Def1; end;
now assume A9: x<>y; p,c,x is_collinear & c,q,y is_collinear by A2,A3,Th1;
then consider z such that A10: p,q,z is_collinear & x,y,z is_collinear by
ANPROJ_2:def 9;
a,b,z is_collinear by A7,A9,A10,ANPROJ_2:def 8; hence thesis by A10,Def1;
end
;
hence thesis by A8; end; hence thesis by A4; end;

theorem
Th12: not a,b,c is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar &
a,b,c,r are_coplanar & a,b,c,s are_coplanar implies p,q,r,s are_coplanar
proof assume A1: not a,b,c is_collinear & a,b,c,p are_coplanar &
a,b,c,q are_coplanar & a,b,c,r are_coplanar & a,b,c,s are_coplanar;
then A2: a,b,p,q are_coplanar & a,b,p,r are_coplanar & a,b,p,s are_coplanar &
a,b,q,s are_coplanar & a,b,r,s are_coplanar & a,b,q,r are_coplanar by Lm4;
A3: a<>b by A1,ANPROJ_2:def 7;
A4: now assume a,b,p is_collinear & a,b,q is_collinear &
a,b,r is_collinear; then p,q,r is_collinear by A3,ANPROJ_2:def 8;
hence thesis by Th10; end;
A5: now assume A6:
not a,b,p is_collinear; then A7: not p,a,b is_collinear by Th1;
p,a,b,q are_coplanar & p,a,b,r are_coplanar & p,a,b,s are_coplanar by A2,Th11
;
then A8: p,a,q,r are_coplanar & p,a,q,s are_coplanar &
p,a,r,s are_coplanar by A7,Lm4; A9: p<>a by A6,ANPROJ_2:def 7;
A10: now assume p,a,q is_collinear & p,a,r is_collinear;
then p,q,r is_collinear by A9,COLLSP:11; hence thesis by Th10; end;
A11: now assume not p,a,q is_collinear; then A12: not p,q,a is_collinear by Th1
; p,q,a,r are_coplanar & p,q,a,s are_coplanar by A8,Th11;
hence thesis by A12,Lm4; end;
now assume not p,a,r is_collinear; then A13: not p,r,a is_collinear by Th1;
p,r,a,q are_coplanar & p,r,a,s are_coplanar by A8,Th11;
then p,r,q,s are_coplanar by A13,Lm4; hence thesis by Th11; end;
hence thesis by A10,A11; end;
A14: now assume A15:
not a,b,q is_collinear; then A16: not q,a,b is_collinear by Th1;
q,a,b,p are_coplanar & q,a,b,r are_coplanar & q,a,b,s are_coplanar by A2,Th11
;
then A17: q,a,p,r are_coplanar & q,a,p,s are_coplanar & q,a,r,s are_coplanar
by A16,Lm4; A18: q<>a by A15,ANPROJ_2:def 7;
A19: now assume q,a,p is_collinear & q,a,r is_collinear;
then q,p,r is_collinear by A18,COLLSP:11; then p,q,r is_collinear by Th1;
hence thesis
by Th10; end;
A20: now assume not q,a,p is_collinear; then A21: not q,p,a is_collinear by Th1
; q,p,a,r are_coplanar & q,p,a,s are_coplanar by A17,Th11;
then q,p,r,s are_coplanar by A21,Lm4; hence thesis by Th11; end;
now assume not q,a,r is_collinear; then A22: not q,r,a is_collinear by Th1;
q,r,a,p are_coplanar & q,r,a,s are_coplanar by A17,Th11;
then q,r,p,s are_coplanar by A22,Lm4; hence thesis by Th11; end;
hence thesis by A19,A20; end;
now assume A23:
not a,b,r is_collinear; then A24: not r,a,b is_collinear by Th1;
r,a,b,p are_coplanar & r,a,b,q are_coplanar & r,a,b,s are_coplanar by A2,Th11
;
then A25: r,a,p,q are_coplanar & r,a,p,s are_coplanar &
r,a,q,s are_coplanar by A24,Lm4; A26: r<>a by A23,ANPROJ_2:def 7;
A27: now assume r,a,p is_collinear & r,a,q is_collinear;
then r,p,q is_collinear by A26,COLLSP:11; then p,q,r is_collinear by Th1;
hence thesis by Th10; end;
A28: now assume not r,a,p is_collinear; then A29: not r,p,a is_collinear by Th1
; r,p,a,q are_coplanar & r,p,a,s are_coplanar by A25,Th11;
then r,p,q,s are_coplanar by A29,Lm4;hence thesis by Th11; end;
now assume not r,a,q is_collinear; then A30: not r,q,a is_collinear by Th1;
r,q,a,p are_coplanar & r,q,a,s are_coplanar by A25,Th11;
then r,q,p,s are_coplanar by A30,Lm4; hence thesis by Th11; end;
hence thesis by A27,A28; end; hence thesis by A4,A5,A14; end;

Lm5: not a,b,c is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar &
a,b,c,r are_coplanar implies a,p,q,r are_coplanar
proof assume A1: not a,b,c is_collinear & a,b,c,p are_coplanar &
a,b,c,q are_coplanar & a,b,c,r are_coplanar;
now assume A2: not p,q,r is_collinear; a<>b by A1,ANPROJ_2:def 7;
then A3:
not a,b,p is_collinear or not a,b,q is_collinear or not a,b,r is_collinear by
A2,ANPROJ_2:def 8;
A4: now assume A5: not a,p,b is_collinear; a,b,p,q are_coplanar &
a,b,p,r are_coplanar by A1,Lm4; then a,p,b,q are_coplanar &
a,p,b,r are_coplanar by Th11; hence thesis by A5,Lm4; end;
A6: now assume A7:not a,q,b is_collinear; a,b,q,p are_coplanar &
a,b,q,r are_coplanar by A1,Lm4; then a,q,b,p are_coplanar &
a,q,b,r are_coplanar by Th11; then a,q,p,r are_coplanar by A7,Lm4;
hence thesis by Th11; end;
now assume A8: not a,r,b is_collinear; a,b,r,p are_coplanar &
a,b,r,q are_coplanar by A1,Lm4; then a,r,b,p are_coplanar &
a,r,b,q are_coplanar by Th11; then a,r,p,q are_coplanar by A8,Lm4;
hence thesis by Th11; end; hence thesis by A3,A4,A6,Th1; end;
hence thesis by Th10; end;

theorem
not p,q,r is_collinear & a,b,c,p are_coplanar & a,b,c,r are_coplanar &
a,b,c,q are_coplanar & p,q,r,s are_coplanar implies a,b,c,s are_coplanar
proof assume A1: not p,q,r is_collinear & a,b,c,p are_coplanar &
a,b,c,r are_coplanar & a,b,c,q are_coplanar & p,q,r,s are_coplanar;
now assume A2: not a,b,c is_collinear; then A3:
a,p,q,r are_coplanar by A1,Lm5;
A4: not b,a,c is_collinear by A2,Th1;
b,b,a is_collinear by ANPROJ_2:def 7;
then A5:b,a,c,b are_coplanar & b,a,c,p are_coplanar & b,a,c,q are_coplanar &
b,a,c,r are_coplanar by A1,Th10,Th11;
A6:not c,a,b is_collinear by A2,Th1; c,a,b,p are_coplanar &
c,a,b,q are_coplanar & c,a,b,r are_coplanar by A1,Th11;
then c,p,q,r are_coplanar by A6,Lm5;
then p,q,r,a are_coplanar & p,q,r,b are_coplanar & p,q,r,c are_coplanar by A3,
A4,A5,Th11,Th12; hence thesis by A1,Th12; end; hence thesis by Th10; end;

Lm6: p<>q & p,q,r is_collinear & a,b,p,q are_coplanar implies
a,b,p,r are_coplanar
proof assume A1: p<>q & p,q,r is_collinear & a,b,p,q are_coplanar;
then consider x such that A2: a,b,x is_collinear & p,q,x is_collinear by Def1;
a,b,x is_collinear & p,r,x is_collinear by A1,A2,COLLSP:11;
hence thesis by Def1; end;

theorem
Th14: p<>q & p,q,r is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar
implies a,b,c,r are_coplanar
proof assume A1: p<>q & p,q,r is_collinear & a,b,c,p are_coplanar &
a,b,c,q are_coplanar;
then A2: a,b,q,c are_coplanar & q,p,r is_collinear by Th1,Th11;
now assume not a,b,c is_collinear;
then a,b,p,q are_coplanar & a,b,q,p are_coplanar by A1,Lm4;
then A3: a,b,p,r are_coplanar & a,b,q,r are_coplanar by A1,A2,Lm6;
A4: now assume a,b,p is_collinear & a,b,q is_collinear; then a,b,r
is_collinear
by A1,COLLSP:14; then r,a,b is_collinear by Th1; hence thesis by Th10; end;
A5: now assume A6:not a,b,p is_collinear; a,a,b is_collinear &
b,a,b is_collinear by ANPROJ_2:def 7
; then a,b,p,a are_coplanar & a,b,p,b are_coplanar &
a,b,p,c are_coplanar by A1,Th10,Th11; hence thesis by A3,A6,Th12; end;
now assume A7:not a,b,q is_collinear; a,a,b is_collinear &
b,a,b is_collinear by ANPROJ_2:def 7
; then a,b,q,a are_coplanar & a,b,q,b are_coplanar &
a,b,q,c are_coplanar by A1,Th10,Th11; hence thesis by A3,A7,Th12; end;
hence thesis by A4,A5; end; hence thesis by Th10; end;

theorem
not a,b,c is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar &
a,b,c,r are_coplanar & a,b,c,s are_coplanar implies ex x st p,q,x is_collinear
& r,s,x is_collinear
proof assume not a,b,c is_collinear & a,b,c,p are_coplanar &
a,b,c,q are_coplanar & a,b,c,r are_coplanar & a,b,c,s are_coplanar;
then p,q,r,s are_coplanar by Th12; then consider x such that A1: p,q,x
is_collinear
& r,s,x is_collinear by Def1; take x; thus thesis by A1; end;

theorem
Th16: ex a,b,c,d st not a,b,c,d are_coplanar
proof consider a,b,c,d such that A1: not a,b,x is_collinear or
not c,d,x is_collinear by ANPROJ_2:def 14
; take a,b,c,d; thus thesis by A1,Def1; end;

theorem
Th17: not p,q,r is_collinear implies ex s st not p,q,r,s are_coplanar
proof assume A1: not p,q,r is_collinear; assume A2: not thesis;
consider a,b,c,d such that A3: not a,b,c,d are_coplanar by Th16;
p,q,r,a are_coplanar & p,q,r,b are_coplanar & p,q,r,c are_coplanar &
p,q,r,d are_coplanar by A2;

theorem
Th18: a=b or a=c or b=c or a=d or b=d or d=c implies a,b,c,d are_coplanar
proof assume A1: a=b or a=c or b=c or a=d or b=d or d=c;
A2: now assume a=b or a=c or b=c; then a,b,c is_collinear by ANPROJ_2:def 7;
hence thesis by Th10; end;
A3: now assume a=d or b=d; then d,a,b is_collinear by ANPROJ_2:def 7;
hence thesis by Th10; end;
now assume d=c; then b,c,d is_collinear by ANPROJ_2:def 7
; hence thesis by Th10; end;
hence thesis by A1,A2,A3; end;

theorem
Th19: not a,b,c,o are_coplanar & o,a,a' is_collinear & a<>a' implies not
a,b,c,a' are_coplanar
proof assume A1: not a,b,c,o are_coplanar & o,a,a' is_collinear & a<>a'; assume
A2: not thesis; a,a',o is_collinear & a,b,c,a are_coplanar by A1,Th1,Th18;

theorem
Th20: not a,b,c is_collinear & not a',b',c' is_collinear &
a,b,c,p are_coplanar & a,b,c,q are_coplanar & a,b,c,r are_coplanar &
a',b',c',p are_coplanar & a',b',c',q are_coplanar & a',b',c',r are_coplanar &
not a,b,c,a' are_coplanar implies p,q,r is_collinear
proof assume A1: not a,b,c is_collinear & not a',b',c' is_collinear &
a,b,c,p are_coplanar & a,b,c,q are_coplanar & a,b,c,r are_coplanar &
a',b',c',p are_coplanar & a',b',c',q are_coplanar & a',b',c',r are_coplanar &
not a,b,c,a' are_coplanar; assume A2: not thesis; a,b,c,a are_coplanar &
a,b,c,b are_coplanar & a,b,c,c are_coplanar by Th18; then A3:
p,q,r,a are_coplanar & p,q,r,b are_coplanar & p,q,r,c are_coplanar by A1,Th12;
a',b',c',a' are_coplanar by Th18; then p,q,r,a' are_coplanar by A1,Th12;

theorem
Th21: a<>a' & o,a,a' is_collinear & not a,b,c,o are_coplanar & not
a',b',c' is_collinear & a,b,p is_collinear & a',b',p is_collinear &
b,c,q is_collinear & b',c',q is_collinear & a,c,r is_collinear &
a',c',r is_collinear implies p,q,r is_collinear
proof assume A1: a<>a' & o,a,a' is_collinear & not a,b,c,o are_coplanar &
not a',b',c' is_collinear & a,b,p is_collinear & a',b',p is_collinear &
b,c,q is_collinear & b',c',q is_collinear & a,c,r is_collinear &
a',c',r is_collinear; then A2: not a,b,c,a' are_coplanar by Th19;
A3: not a,b,c is_collinear by A1,Th10; p,a,b is_collinear by A1,Th1;
then A4: a,b,c,p are_coplanar by Th10; A5: a,b,c,q are_coplanar by A1,Th10;
c,r,a is_collinear by A1,Th1; then A6: a,b,c,r are_coplanar by Th10;
p,a',b' is_collinear by A1,Th1; then A7: a',b',c',p are_coplanar by Th10;
A8: a',b',c',q are_coplanar by A1,Th10; c',r,a' is_collinear by A1,Th1;
then a',b',c',r are_coplanar by Th10;
hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,Th20; end;

theorem
Th22: not a,b,c,d are_coplanar & a,b,c,o are_coplanar & not a,b,o is_collinear
implies not a,b,d,o are_coplanar
proof assume A1: not a,b,c,d are_coplanar & a,b,c,o are_coplanar &
not a,b,o is_collinear; assume not thesis; then A2: a,b,o,d are_coplanar by
Th11; a,b,o,c are_coplanar by A1,Th11;

theorem
Th23: not a,b,c,o are_coplanar & o,a,a' is_collinear & o,b,b' is_collinear
& o,c,c' is_collinear & o<>a' & o<>b' & o<>c' implies
not a',b',c' is_collinear & not a',b',c',o are_coplanar
proof assume A1: not a,b,c,o are_coplanar & o,a,a' is_collinear &
o,b,b' is_collinear & o,c,c' is_collinear & o<>a' & o<>b' & o<>c';
then a,o,a' is_collinear & not o,b,c,a are_coplanar by Th1,Th11;
then not o,b,c,a' are_coplanar by A1,Th19; then not o,a',b,c are_coplanar &
c,o,c' is_collinear by A1,Th1,Th11; then not o,a',b,c' are_coplanar by A1,Th19
;
then not o,a',c',b are_coplanar & b,o,b' is_collinear by A1,Th1,Th11;
then not o,a',c',b' are_coplanar by A1,Th19; then not a',b',c',o
are_coplanar by Th11; hence thesis by Th10; end;

theorem
Th24: a,b,c,o are_coplanar & not a,b,c,d are_coplanar & not
a,b,d,o are_coplanar & not b,c,d,o are_coplanar & not a,c,d,o are_coplanar
& o,d,d' is_collinear & o,a,a' is_collinear & o,b,b' is_collinear &
o,c,c' is_collinear & a,d,s is_collinear & a',d',s is_collinear &
b,d,t is_collinear & b',d',t is_collinear & c,d,u is_collinear & o<>a' & o<>d'
& d<>d' & o<>b' implies not s,t,u is_collinear
proof assume A1: a,b,c,o are_coplanar & not a,b,c,d are_coplanar &
not a,b,d,o are_coplanar & not b,c,d,o are_coplanar & not a,c,d,o are_coplanar
& o,d,d' is_collinear & o,a,a' is_collinear & o,b,b' is_collinear &
o,c,c' is_collinear & a,d,s is_collinear & a',d',s is_collinear &
b,d,t is_collinear & b',d',t is_collinear & c,d,u is_collinear & o<>a' & o<>d'
& d<>d' & o<>b'; then not d,o,a is_collinear by Th10;
then not o,a,d is_collinear by Th1; then A2: s<>d by A1,Th8;
A3:not d,b,c,a are_coplanar by A1,Th11; then A4: not d,b,c,s are_coplanar by A1
,A2,Th19; A5: d,b,c,b are_coplanar & d,b,c,d are_coplanar &
d,b,c,c are_coplanar by Th18; b<>d by A1,Th18;
then A6: d,b,c,t are_coplanar by A1,A5,Th14;
A7: d,b,c,u are_coplanar by A1,A5,Th14; A8: not d,b,c is_collinear by A3,Th10;
not b,d,o is_collinear by A1,Th10; then not o,b,d is_collinear by Th1;
then A9: t<>d by A1,Th8; A10: d,b,t is_collinear by A1,Th1; d,c,u
is_collinear
by A1,Th1; then t<>u by A8,A9,A10,Lm1; then not t,u,s is_collinear by A4,A6,A7,
Th14; hence thesis by Th1; end;

definition let FCPS,o,a,b,c;
:Def2: not a,b,c is_collinear & not o,a,b is_collinear & not o,b,c is_collinear
& not o,c,a is_collinear;
end;

Lm7: o<>a' & o<>b' & o<>c' & a<>a' & b<>b' & o,a,b,c constitute_a_quadrangle
& o,a,a' is_collinear & o,b,b' is_collinear & o,c,c' is_collinear &
a,b,p is_collinear & a',b',p is_collinear & b,c,q is_collinear &
b',c',q is_collinear & a,c,r is_collinear & a',c',r is_collinear
implies p,q,r is_collinear
proof assume that A1: o<>a' & o<>b' & o<>c' & a<>a' & b<>b' and
A3: o,a,a' is_collinear & o,b,b' is_collinear &
o,c,c' is_collinear & a,b,p is_collinear & a',b',p is_collinear &
b,c,q is_collinear & b',c',q is_collinear & a,c,r is_collinear &
a',c',r is_collinear;
A4: o<>a' & o<>b' & o<>c' & a<>a' & b<>b' &
not o,a,b is_collinear & not o,b,c is_collinear & not o,c,a is_collinear &
not a,b,c is_collinear & o,a,a' is_collinear & o,b,b' is_collinear &
o,c,c' is_collinear & a,b,p is_collinear & a',b',p is_collinear &
b,c,q is_collinear & b',c',q is_collinear & a,c,r is_collinear &
a',c',r is_collinear by A1,A2,A3,Def2;
A5: now assume A6: not a,b,c,o are_coplanar; then not a',b',c' is_collinear
by A1,A3,Th23; hence thesis by A1,A3,A6,Th21; end;
now assume A7: a,b,c,o are_coplanar; consider d such that
A8: not a,b,c,d are_coplanar by A4,Th17; consider d' such that A9: o<>d' &
d<>d' & o,d,d' is_collinear by ANPROJ_2:def 10; A10: a,o,a' is_collinear &
b,o,b' is_collinear & c,o,c' is_collinear by A3,Th1; then consider s such that
A11: a,d,s is_collinear & a',d',s is_collinear by A9,ANPROJ_2:def 9;
consider t such that A12: b,d,t is_collinear & b',d',t is_collinear by A9,A10,
ANPROJ_2:def 9
; consider u such that A13: c,d,u is_collinear & c',d',u is_collinear
by A9,A10,ANPROJ_2:def 9; A14: not a,b,o is_collinear & not b,c,o is_collinear
&
not a,c,o is_collinear by A4,Th1;
then A15: not a,b,d,o are_coplanar by A7,A8,Th22; not b,c,a,d are_coplanar &
b,c,a,o are_coplanar by A7,A8,Th11; then A16: not b,c,d,o are_coplanar by A14,
Th22; not a,c,b,d are_coplanar & a,c,b,o are_coplanar by A7,A8,Th11;
then A17: not a,c,d,o are_coplanar by A14,Th22; then A18: not s,t,u
is_collinear
by A1,A3,A7,A8,A9,A11,A12,A13,A15,A16,Th24;
not a',b',d' is_collinear by A1,A3,A9,A15,Th23; then p,t,s is_collinear by A1
,A3,A11,A12,A15,Th21; then A19: t,s,p is_collinear by Th1; not b',c',d'
is_collinear
by A1,A3,A9,A16,Th23; then q,u,t is_collinear by A1,A3,A12,A13,A16,Th21;
then A20:
u,t,q is_collinear by Th1; not a',c',d' is_collinear by A1,A3,A9,A17,Th23;
then r,u,s is_collinear by A1,A3,A11,A13,A17,Th21; then A21: s,u,r
is_collinear by Th1;
A22: d,a,s is_collinear & d',a',s is_collinear by A11,Th1;
not d,o,a is_collinear by A15,Th10; then not o,d,a is_collinear by Th1;
then s<>a by A1,A3,A9,A22,Th8; then A23: not a,b,c,s are_coplanar by A8,A22,
Th19;
A24: a<>b & b<>c & c <>a & s<>t & t<>u & u<>s by A4,A18,ANPROJ_2:def 7;
A25: a,b,c,a are_coplanar & a,b,c,b are_coplanar & a,b,c,c are_coplanar &
s,t,u,s are_coplanar & s,t,u,t are_coplanar & s,t,u,u are_coplanar by Th18;
then A26: a,b,c,p are_coplanar & a,b,c,q are_coplanar & a,b,c,r are_coplanar
by A3,A24,Th14;
s,t,u,p are_coplanar & s,t,u,q are_coplanar & s,t,u,r are_coplanar by A19,
A20,A21,A24,A25,Th14;
hence thesis by A4,A18,A23,A26,Th20; end; hence thesis by A5; end;

canceled;

theorem
Th26: not o,a,b is_collinear & not o,b,c is_collinear & not o,a,c is_collinear
& o,a,a' is_collinear & o,b,b' is_collinear & o,c,c' is_collinear &
a,b,p is_collinear & a',b',p is_collinear & a<>a' & b,c,r is_collinear &
b',c',r is_collinear & a,c,q is_collinear & b<>b' & a',c',q is_collinear &
o<>a' & o<>b' & o<>c' implies r,q,p is_collinear
proof assume A1: not o,a,b is_collinear & not o,b,c is_collinear &
not o,a,c is_collinear & o,a,a' is_collinear & o,b,b' is_collinear &
o,c,c' is_collinear & a,b,p is_collinear & a',b',p is_collinear & a<>a' &
b,c,r is_collinear & b',c',r is_collinear & a,c,q is_collinear & b<>b' &
a',c',q is_collinear & o<>a' & o<>b' & o<>c';
then A2: not o,c,a is_collinear by Th1;
A3: now assume not a,b,c is_collinear;
then p,r,q is_collinear by A1,Lm7; hence thesis by Th1; end;
now assume A4: a,b,c is_collinear; A5: a<>b & b<>c & c <>a
by A1,ANPROJ_2:def 7;
b,c,a is_collinear & b,c,b is_collinear by A4,Th1,ANPROJ_2:def 7;
then A6: a,b,r is_collinear by A1,A5,ANPROJ_2:def 8;
a,c,a is_collinear & a,c,b is_collinear by A4,Th1,ANPROJ_2:def 7;
then a,b,q is_collinear by A1,A5,ANPROJ_2:def 8; hence thesis
by A1,A5,A6,ANPROJ_2:def 8; end;
hence thesis by A3; end;

theorem Th27:
for CS being up-3-dimensional CollProjectiveSpace holds
CS is Desarguesian
proof let CS be up-3-dimensional CollProjectiveSpace;
for o,a,b,c,a',b',c',r,q,p being Element of CS st
o<>a' & a<>a' & o<>b' & b<>b' & o<>c' & c <>c' &
not o,a,b is_collinear & not o,a,c is_collinear &
not o,b,c is_collinear & a,b,p is_collinear & a',b',p is_collinear &
b,c,r is_collinear & b',c',r is_collinear & a,c,q is_collinear &
a',c',q is_collinear & o,a,a' is_collinear & o,b,b' is_collinear &
o,c,c' is_collinear holds r,q,p is_collinear by Th26;
hence thesis by ANPROJ_2:def 12; end;

definition
cluster -> Desarguesian (up-3-dimensional CollProjectiveSpace);
correctness by Th27;
end;
```