
theorem Th30:
  for P being Element of absolute holds tangent P misses BK_model
  proof
    let P be Element of absolute;
    assume not tangent P misses BK_model;
    then consider x be object such that
A1: x in tangent P and
A2: x in BK_model by XBOOLE_0:3;
    reconsider x as Element of real_projective_plane by A1;
    reconsider L = tangent P as LINE of IncProjSp_of real_projective_plane
      by INCPROJ:4;
    reconsider ip = P,iq = x as POINT of IncProjSp_of real_projective_plane
      by INCPROJ:3;
    P in tangent P & x in tangent P by A1,Th21;
    then ip on L & iq 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
A3: p1 = P1 and
A4: p2 = P2 and
A5: P1 <> P2 and
A6: P1 in absolute and
A7: P2 in absolute and
A8: p1 on L and
A9: p2 on L by A2,Th15;
    P1 in L & P2 in L by A3,A4,A8,A9,INCPROJ:5;
    then P1 in tangent P /\ absolute & P2 in tangent P /\ absolute
      by A6,A7,XBOOLE_0:def 4;
    then P1 in {P} & P2 in {P} by Th22;
    then P1 = P & P2 = P by TARSKI:def 1;
    hence contradiction by A5;
  end;
