reserve P for Element of BK_model;

theorem Th12:
  for L being LINE of IncProjSp_of real_projective_plane
  st P in L holds ex P1,P2 being Element of absolute st P1 <> P2 &
  P1 in L & P2 in L
  proof
    let L be LINE of IncProjSp_of real_projective_plane;
    assume
A1: P in L;
    consider Q be Element of ProjectiveSpace TOP-REAL 3 such that
A2: P <> Q and
A3: Q in L by BKMODEL2:8;
    consider R be Element of absolute such that
A4: P,Q,R are_collinear by A2,BKMODEL2:19;
    reconsider p = P,r = R as POINT of IncProjSp_of real_projective_plane
      by INCPROJ:3;
    reconsider L9 = L as LINE of real_projective_plane by INCPROJ:4;
    Line(P,Q) = L9 by A1,A2,A3,COLLSP:19;
    then P in L9 & Q in L9 & R in L9 by A1,A3,A4,COLLSP:11;
    then p on L & r 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
A5: p1 = P1 and
A6: p2 = P2 and
A7: P1 <> P2 and
A8: P1 in absolute and
A9: P2 in absolute and
A10:p1 on L and
A11:p2 on L by BKMODEL2:23;
    reconsider P1,P2 as Element of absolute by A8,A9;
    take P1,P2;
    P1 in L9 & P2 in L9 by A5,A6,A10,A11,INCPROJ:5;
    hence thesis by A7;
  end;
