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 Th13:
  not p1,p2,q1 are_collinear & p1,p2,p3 are_collinear & q1,q2,p1
  are_collinear & p1<>p3 & p1<>q2 implies not p3,p1,q2 are_collinear
proof
  assume that
A1: not p1,p2,q1 are_collinear and
A2: p1,p2,p3 are_collinear and
A3: q1,q2,p1 are_collinear and
A4: p1<>p3 and
A5: p1<>q2;
A6: p1,q2,q1 are_collinear by A3,Th1;
  assume not thesis;
  then
A7: p1,p3,q2 are_collinear by Th1;
  p1,p3,p2 are_collinear by A2,Th1;
  then p1,q2,p2 are_collinear by A4,A7,Th2;
  hence contradiction by A1,A5,A6,Th2;
end;
