
theorem Th15:
  for L being LINE of IncProjSp_of real_projective_plane
  for p ,q being POINT of IncProjSp_of real_projective_plane
  for P,Q being Element of real_projective_plane st
  p = P & q = Q &
  P in BK_model & Q in absolute &
  q on L & p on L holds
  ex p1,p2 being POINT of IncProjSp_of real_projective_plane,
  P1,P2 being Element of real_projective_plane st
  p1 = P1 & p2 = P2 & P1 <> P2 &
  P1 in absolute & P2 in absolute &
  p1 on L & p2 on L
  proof
    let L being LINE of IncProjSp_of real_projective_plane;
    let p ,q being POINT of IncProjSp_of real_projective_plane;
    let P ,Q being Element of real_projective_plane;
    assume that
A1: p = P and
A2: q = Q and
A3: P in BK_model and
A4: Q in absolute and
A5: q on L and
A6: p on L;
A7: P <> Q by Th01,A3,A4,XBOOLE_0:def 4;
    reconsider l = L as LINE of real_projective_plane by INCPROJ:4;
A8: P in l by A1,A6,INCPROJ:5;
    reconsider PBK = P as Element of BK_model by A3;
    consider R being Element of real_projective_plane such that
A9: R in BK_model and
A10: P <> R and
A11: R,P,Q are_collinear by A3,A4,Th14;
    reconsider r = R as POINT of IncProjSp_of real_projective_plane
      by INCPROJ:3;
    consider LL be LINE of IncProjSp_of real_projective_plane such that
A12: r on LL & p on LL & q on LL by A1,A2,A11,INCPROJ:10;
    L = LL by A1,A2,A5,A6,A12,A7,INCPROJ:8;
    then R in l by A12,INCPROJ:5; then
A13: l = Line(P,R) by A8,A10,COLLSP:19;
    reconsider RBK = R as Element of BK_model by A9;
    consider P1,P2 be Element of absolute such that
A14: P1 <> P2 and
A15: PBK,RBK,P1 are_collinear and
A16: PBK,RBK,P2 are_collinear by A10,Th12;
     reconsider PP1 = P1, PP2 = P2 as Element of real_projective_plane;
A17: PP1 in Line(P,R) & PP2 in Line(P,R) by A15,A16,COLLSP:11;
    reconsider p1 = P1,p2 = P2 as POINT of IncProjSp_of real_projective_plane
      by INCPROJ:3;
    p1 on L & p2 on L by A13,A17,INCPROJ:5;
    hence thesis by A14;
  end;
