
theorem
  for P,Q being Element of BK_model for P1,P2,P3 being Element of absolute st
  P <> Q & P1 <> P2 & P,Q,P1 are_collinear & P,Q,P2 are_collinear &
  P,Q,P3 are_collinear holds P3 = P1 or P3 = P2
  proof
    let P,Q being Element of BK_model;
    let P1,P2,P3 being Element of absolute;
    assume that
A1: P <> Q and
A2: P1 <> P2 and
A3: P,Q,P1 are_collinear and
A4: P,Q,P2 are_collinear and
A5: P,Q,P3 are_collinear;
    P3 = P1 or P3 = P2
    proof
      assume P3 <> P1 & P3 <> P2;
      then P1,P2,P3 are_mutually_distinct by A2;
      hence contradiction by A1,A3,A4,A5,COLLSP:3,BKMODEL1:92;
    end;
    hence thesis;
  end;
