reserve CPS for proper CollSp,
  B for Subset of CPS,
  p for Point of CPS,
  x, y, z, Y for set;
reserve a,b,c,p,q for POINT of IncProjSp_of(CPS),
  P,Q for LINE of IncProjSp_of(CPS),
  a9,b9,c9,p9,q9,r9 for Point of CPS,
  P9 for LINE of CPS;
reserve CPS for CollProjectiveSpace,
  a,b,c,d,p,q for POINT of IncProjSp_of(CPS ),
  P,Q,S,M,N for LINE of IncProjSp_of(CPS),
  a9,b9,c9,d9,p9,q9 for Point of CPS;

theorem Th13:
  a on M & b on M & c on N & d on N & p on M & p on N & a on P & c
on P & b on Q & d on Q & not p on P & not p on Q & M<>N implies ex q st q on P
  & q on Q
proof
  assume that
A1: a on M and
A2: b on M and
A3: c on N and
A4: d on N and
A5: p on M & p on N and
A6: a on P and
A7: c on P and
A8: b on Q and
A9: d on Q and
A10: not p on P and
A11: not p on Q and
A12: M<>N;
  reconsider a9=a,b9=b,c9=c,d9=d,p9=p as Point of CPS;
  b9,p9,a9 are_collinear & p9,d9,c9 are_collinear by A1,A2,A3,A4,A5,Th10;
  then consider q9 such that
A13: b9,d9,q9 are_collinear and
A14: a9,c9,q9 are_collinear by ANPROJ_2:def 9;
  reconsider q = q9 as POINT of IncProjSp_of(CPS);
A15: ex P2 being LINE of IncProjSp_of(CPS) st b on P2 & d on P2 & q on P2 by
A13,Th10;
  b<>d by A2,A4,A5,A8,A11,A12,Th8;
  then
A16: q on Q by A8,A9,A15,Th8;
A17: ex P1 being LINE of IncProjSp_of(CPS) st a on P1 & c on P1 & q on P1 by
A14,Th10;
  a<>c by A1,A3,A5,A6,A10,A12,Th8;
  then q on P by A6,A7,A17,Th8;
  hence thesis by A16;
end;
