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 Th15:
  (ex p, p1, r, r1 being Point of CPS st not ex s being Point of
CPS st (p, p1, s are_collinear & r, r1, s are_collinear)) implies
ex M, N st not
  ex q st q on M & q on N
proof
  given p, p1, r, r1 being Point of CPS such that
A1: not ex s being Point of CPS st
(p,p1,s are_collinear & r,r1,s are_collinear);
  set M99= Line(p,p1), N99= Line(r,r1);
  p<>p1 & r<>r1
  proof
A2: now
      assume p = p1;
      then
A3:   p,p1,r1 are_collinear by COLLSP:2;
      r,r1,r1 are_collinear by COLLSP:2;
      hence contradiction by A1,A3;
    end;
A4: now
      assume r = r1;
      then p,p1,p1 are_collinear & r,r1,p1 are_collinear by COLLSP:2;
      hence contradiction by A1;
    end;
    assume not thesis;
    hence contradiction by A2,A4;
  end;
  then reconsider M9= M99, N9= N99 as LINE of CPS by COLLSP:def 7;
  reconsider M = M9, N = N9 as LINE of IncProjSp_of(CPS) by Th1;
  take M, N;
  thus not ex q st q on M & q on N
  proof
    assume not thesis;
    then consider q such that
A5: q on M and
A6: q on N;
    reconsider q9=q as Point of CPS;
    q9 in N99 by A6,Th5;
    then
A7: r,r1,q9 are_collinear by COLLSP:11;
    q9 in M99 by A5,Th5;
    then p,p1,q9 are_collinear by COLLSP:11;
    hence contradiction by A1,A7;
  end;
end;
