
theorem Th59:
  for P,Q,R being Point of real_projective_plane holds
  P,Q,R are_collinear iff dual P,dual Q,dual R are_concurrent
  proof
    let P,Q,R be Point of real_projective_plane;
    thus P,Q,R are_collinear implies dual P, dual Q, dual R are_concurrent
      by Th56;
    assume dual P, dual Q, dual R are_concurrent;
    then dual dual P, dual dual Q, dual dual R are_collinear
      by Th58;
    then P, dual dual Q, dual dual R are_collinear by Th45;
    then P, Q, dual dual R are_collinear by Th45;
    hence thesis by Th45;
  end;
