reserve PCPP for CollProjectiveSpace,
  a,a9,a1,a2,a3,b,b9,b1,b2,c,c1,c3,d,d9,e,
  o,p,p1,p2,p3,r,q, q1,q2,q3,x,y for Element of PCPP;

theorem Th14:
  a1<>a2 & b1<>b2 & b1,b2,x are_collinear & b1,b2,y are_collinear &
a1,a2,x are_collinear & a1,a2,y are_collinear & not a1,a2,b1 are_collinear
implies
  x=y
proof
  assume that
A1: a1<>a2 and
A2: b1<>b2 & b1,b2,x are_collinear & b1,b2,y are_collinear and
A3: a1,a2,x are_collinear & a1,a2,y are_collinear and
A4: not a1,a2,b1 are_collinear;
  a1,a2,a2 are_collinear by ANPROJ_2:def 7;
  then
A5: x,y,a2 are_collinear by A1,A3,ANPROJ_2:def 8;
  b1,b2,b1 are_collinear by ANPROJ_2:def 7;
  then
A6: x,y,b1 are_collinear by A2,ANPROJ_2:def 8;
  assume
A7: not thesis;
  a1,a2,a1 are_collinear by ANPROJ_2:def 7;
  then x,y,a1 are_collinear by A1,A3,ANPROJ_2:def 8;
  hence contradiction by A4,A7,A6,A5,ANPROJ_2:def 8;
end;
