theorem
  not a,b // c,d & a,b,p1 are_collinear & a,b,p2 are_collinear & c,d,p1
  are_collinear & c,d,p2 are_collinear implies p1=p2
proof
  assume that
A1: not a,b // c,d and
A2: a,b,p1 are_collinear & a,b,p2 are_collinear and
A3: c,d,p1 are_collinear & c,d,p2 are_collinear;
  c,d // p1,p2 by A3,Th26;
  then
A4: p1,p2 // c,d by Th6;
  a,b // p1,p2 by A2,Th26;
  then p1,p2 // a,b by Th6;
  hence thesis by A1,A4,Def1;
end;
