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 Th16:
  (for p, p1, q, q1, r2 being Point of CPS ex r, r1 being Point of
  CPS st p,q,r are_collinear & p1,q1,r1 are_collinear & r2,r,r1 are_collinear)
  implies for a, M, N ex b, c, S st a on S & b on S & c on S & b on M & c on N
proof
  assume
A1: for p, p1, q, q1, r2 being Point of CPS ex r, r1 being Point of CPS
  st p,q,r are_collinear & p1,q1,r1 are_collinear & r2,r,r1 are_collinear;
  let a, M, N;
  reconsider M9= M, N9= N as LINE of CPS by Th1;
  reconsider a9= a as Point of CPS;
  consider p, q being Point of CPS such that
  p<>q and
A2: M9= Line(p,q) by COLLSP:def 7;
  consider p1, q1 being Point of CPS such that
  p1<>q1 and
A3: N9 = Line(p1,q1) by COLLSP:def 7;
  consider b9, c9 such that
A4: p,q,b9 are_collinear and
A5: p1,q1,c9 are_collinear and
A6: a9,b9,c9 are_collinear by A1;
  reconsider b = b9, c = c9 as POINT of IncProjSp_of(CPS);
  b9 in M9 by A2,A4,COLLSP:11;
  then
A7: b on M by Th5;
  c9 in N9 by A3,A5,COLLSP:11;
  then
A8: c on N by Th5;
  ex S st a on S & b on S & c on S by A6,Th10;
  hence thesis by A7,A8;
end;
