
theorem Th56:
  for P,Q,P1,P2 being non point_at_infty Element of ProjectiveSpace
  TOP-REAL 3 st P <> Q & P in BK_model & Q in BK_model & P1 in absolute &
  P2 in absolute & P1 <> P2 & P,Q,P1 are_collinear & P,Q,P2 are_collinear
  holds between RP3_to_T2 Q,RP3_to_T2 P,RP3_to_T2 P1 or
  between RP3_to_T2 Q,RP3_to_T2 P,RP3_to_T2 P2
  proof
    let P,Q,P1,P2 be non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    assume that
A1: P <> Q and
A2: P in BK_model and
A3: Q in BK_model and
A4: P1 in absolute and
A5: P2 in absolute and
A6: P1 <> P2 and
A7: P,Q,P1 are_collinear and
A8: P,Q,P2 are_collinear;
    set  P9 = RP3_to_T2 P, Q9 = RP3_to_T2 Q, P19 = RP3_to_T2 P1,
        P29 = RP3_to_T2 P2;
    Collinear P9,Q9,P19 & Collinear P9,Q9,P29 &
      not between Q9,P19,P9 & not between Q9,P29,P9
      by A2,A3,A4,A5,A7,A8,Th47,Th46;
    hence thesis by A1,A2,A4,A5,A6,Th55,GTARSKI1:16;
  end;
