
theorem Th56:
  for P,Q,R being Point of real_projective_plane st P,Q,R are_collinear holds
  dual P,dual Q, dual R are_concurrent
  proof
    let P,Q,R be Point of real_projective_plane;
    assume
A1: P,Q,R are_collinear;
    per cases;
    suppose
A2:  Q = R;
      reconsider lP = dual P,lQ = dual Q as LINE of real_projective_plane
        by INCPROJ:1;
      ex x be object st x in lP & x in lQ by BKMODEL1:76,XBOOLE_0:3;
      hence thesis by A2;
    end;
    suppose
A3:   Q <> R;
A4:  Q,R,P are_collinear by A1,ANPROJ_2:24;
      reconsider l = Line(Q,R) as LINE of real_projective_plane
        by A3,COLLSP:def 7;
      l in {B where B is Subset of real_projective_plane:
        B is LINE of real_projective_plane};
      then reconsider l = Line(Q,R) as Element of ProjectiveLines
        real_projective_plane;
      dual l in dual P & dual l in dual Q & dual l in dual R
        by A4,COLLSP:11,Th54,COLLSP:10;
      hence thesis;
    end;
  end;
