
theorem Th16:
  for P being Element of BK_model
  for Q being Element of absolute holds
  ex R being Element of absolute st Q <> R & Q,P,R are_collinear
  proof
    let P be Element of BK_model;
    let Q be Element of absolute;
A1: P <> Q by XBOOLE_0:def 4,Th01;
    reconsider p9 = P,q9 = Q as Element of real_projective_plane;
    reconsider L9 = Line(p9,q9) as LINE of real_projective_plane
      by A1,COLLSP:def 7;
    reconsider L = L9 as LINE of IncProjSp_of real_projective_plane
      by INCPROJ:4;
    reconsider p = P,q = Q as POINT of IncProjSp_of real_projective_plane
      by INCPROJ:3;
    p9 in L9 & q9 in L9 by COLLSP:10;
    then p on L & q on L by INCPROJ:5;
    then consider p1,p2 be POINT of IncProjSp_of real_projective_plane,
    P1,P2 be Element of real_projective_plane such that
A2: p1 = P1 & p2 = P2 & P1 <> P2 &
    P1 in absolute & P2 in absolute &
    p1 on L & p2 on L by Th15;
    reconsider p1,p2 as Element of real_projective_plane by INCPROJ:3;
A3: P1 in L9 & P2 in L9 by A2,INCPROJ:5;
    then
A4: p9,q9,p1 are_collinear & p9,q9,p2 are_collinear & P1 <> P2 &
      P1 in absolute & P2 in absolute by A2,COLLSP:11;
    reconsider P1,P2 as Element of absolute by A2;
    per cases;
    suppose
A5:   q9 = p1;
      take P2;
      now
        thus Q <> P2 by A5,A2;
        P,Q,P2 are_collinear by A3,COLLSP:11;
        hence Q,P,P2 are_collinear by COLLSP:4;
      end;
      hence thesis;
    end;
    suppose q9 <> p1;
      per cases;
      suppose
A6:     Q <> P2;
        take P2;
        P,Q,P2 are_collinear by A3,COLLSP:11;
        hence thesis by A6,COLLSP:4;
      end;
      suppose
A7:     Q = P2;
        take P1;
        thus thesis by A4,A7,A2,COLLSP:4;
      end;
    end;
  end;
