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 Th9:
  ex P st p on P & q on P
proof
  reconsider p9= p,q9= q as Point of CPS;
A1: now
    consider r9 such that
A2: p9<>r9 by Th7;
    consider P9 being LINE of CPS such that
A3: p9 in P9 and
    r9 in P9 by A2,COLLSP:15;
    reconsider P = P9 as LINE of IncProjSp_of(CPS) by Th1;
    assume
A4: p9=q9;
    p on P by A3,Th5;
    hence thesis by A4;
  end;
  now
    assume p9<>q9;
    then consider P9 being LINE of CPS such that
A5: p9 in P9 & q9 in P9 by COLLSP:15;
    reconsider P = P9 as LINE of IncProjSp_of(CPS) by Th1;
    p on P & q on P by A5,Th5;
    hence thesis;
  end;
  hence thesis by A1;
end;
