reserve FCPS for up-3-dimensional CollProjectiveSpace;
reserve a,a9,b,b9,c,c9,d,d9,o,p,q,r,s,t,u,x,y,z for Element of FCPS;

theorem Th22:
  for CS being up-3-dimensional CollProjectiveSpace holds CS is Desarguesian
proof
  let CS be up-3-dimensional CollProjectiveSpace;
  for o,a,b,c,a9,b9,c9,r,q,p being Element of CS st o<>a9 & a<>a9 & o<>b9
  & b<>b9 & o<>c9 & c <>c9 & not o,a,b are_collinear &
not o,a,c are_collinear &
  not o,b,c are_collinear & a,b,p are_collinear & a9,b9,p are_collinear
& b,c,r
are_collinear & b9,c9,r are_collinear & a,c,q are_collinear &
a9,c9,q are_collinear
  & o,a,a9 are_collinear & o,b,b9 are_collinear & o,c,c9 are_collinear
holds r,q,p
  are_collinear by Th21;
  hence thesis by ANPROJ_2:def 12;
end;
