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 Th8:
  not o,a,d are_collinear & o,d,d9 are_collinear & d<>d9 & a9,d9,x
  are_collinear & o,a,a9 are_collinear & o<>a9 implies x<>d
proof
  assume that
A1: not o,a,d are_collinear and
A2: o,d,d9 are_collinear and
A3: d<>d9 and
A4: a9,d9,x are_collinear and
A5: o,a,a9 are_collinear and
A6: o<>a9;
  assume not thesis;
  then
A7: d,d9,a9 are_collinear by A4,Th1;
  d,d9,o are_collinear by A2,Th1;
  then d,o,a9 are_collinear by A3,A7,Th2;
  then
A8: o,a9,d are_collinear by Th1;
  o,a9,a are_collinear by A5,Th1;
  hence contradiction by A1,A6,A8,Th2;
end;
