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 Th15:
  not o,a1,a2 are_collinear & o,a1,b1 are_collinear & o,a2,b2
  are_collinear & o<>b1 & o<>b2 implies not o,b1,b2 are_collinear
proof
  assume that
A1: ( not o,a1,a2 are_collinear)& o,a1,b1 are_collinear and
A2: o,a2,b2 are_collinear and
A3: o<>b1 and
A4: o<>b2;
  not o,b1,a2 are_collinear by A1,A3,Th6;
  then not o,a2,b1 are_collinear by Th1;
  then not o,b2,b1 are_collinear by A2,A4,Th6;
  hence thesis by Th1;
end;
