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 Th14:
  (for a9,b9,c9,d9 ex p9 st a9, b9, p9 are_collinear & c9, d9, p9
  are_collinear) implies for M, N ex q st q on M & q on N
proof
  assume
A1: for a9,b9,c9,d9 ex p9 st a9, b9, p9 are_collinear &
c9, d9, p9 are_collinear;
  let M, N;
  reconsider M9= M, N9= N as LINE of CPS by Th1;
  consider a9, b9 such that
  a9<>b9 and
A2: M9= Line(a9,b9) by COLLSP:def 7;
  consider c9, d9 such that
  c9<>d9 and
A3: N9= Line(c9,d9) by COLLSP:def 7;
  consider p9 such that
A4: a9, b9, p9 are_collinear and
A5: c9, d9, p9 are_collinear by A1;
  reconsider q = p9 as POINT of IncProjSp_of(CPS);
  p9 in N9 by A3,A5,COLLSP:11;
  then
A6: q on N by Th5;
  p9 in M9 by A2,A4,COLLSP:11;
  then q on M by Th5;
  hence thesis by A6;
end;
