
theorem Th32:
  for P,R,S being Element of real_projective_plane
  for Q being Element of absolute st P in BK_model & R in tangent Q &
  P,S,R are_collinear & R <> S holds Q <> S
  proof
    let P,R,S being Element of real_projective_plane;
    let Q being Element of absolute;
    assume that
A1: P in BK_model and
A2: R in tangent Q and
A3: P,S,R are_collinear and
A4: R <> S;
A5: S,R,P are_collinear by A3,COLLSP:8;
    consider q be Element of real_projective_plane such that
A6: q = Q & tangent Q = Line(q,pole_infty Q) by Def04;
    assume Q = S;
    then q,pole_infty Q,S are_collinear & q,pole_infty Q,R are_collinear
      by A2,Th21,A6,COLLSP:11;
    then
A7: P in tangent Q by A5,A4,COLLSP:9,A6,COLLSP:11;
    reconsider L = tangent Q as LINE of IncProjSp_of real_projective_plane
      by INCPROJ:4;
    reconsider ip = P,iq = Q as POINT of IncProjSp_of real_projective_plane
      by INCPROJ:3;
    Q in tangent Q by Th21;
    then ip on L & iq on L by A7,INCPROJ:5;
    then consider p1,p2 be POINT of IncProjSp_of real_projective_plane,
                  P1,P2 be Element of real_projective_plane such that
A8: p1 = P1 & p2 = P2 & P1 <> P2 & P1 in absolute & P2 in absolute &
    p1 on L & p2 on L by A1,Th15;
    P1 in L & P2 in L by A8,INCPROJ:5;
    then P1 in tangent Q /\ absolute & P2 in tangent Q /\ absolute
      by A8,XBOOLE_0:def 4;
    then P1 in {Q} & P2 in {Q} by Th22;
    then P1 = Q & P2 = Q by TARSKI:def 1;
    hence contradiction by A8;
  end;
