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 Th4:
  p<>q implies ex r st not p,q,r are_collinear
proof
  consider a,b,c such that
A1: not a,b,c are_collinear by COLLSP:def 6;
  assume p<>q;
  then p,q,a are_collinear & p,q,b are_collinear & p,q,c are_collinear implies
  contradiction by A1,ANPROJ_2:def 8;
  hence thesis;
end;
