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;

theorem Th11:
  ex p, P st not p on P
proof
  consider p9,q9,r9 such that
A1: not p9,q9,r9 are_collinear by COLLSP:def 6;
  set X = Line(p9,q9);
  p9 <> q9 by A1,COLLSP:2;
  then reconsider P9= X as LINE of CPS by COLLSP:def 7;
  reconsider P = P9 as LINE of IncProjSp_of(CPS) by Th1;
  reconsider r = r9 as POINT of IncProjSp_of(CPS);
  not r on P
  proof
    assume not thesis;
    then r9 in X by Th5;
    hence contradiction by A1,COLLSP:11;
  end;
  hence thesis;
end;
