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 Th10:
  a = a9 & b = b9 & c = c9 implies (a9,b9,c9 are_collinear iff ex P
  st a on P & b on P & c on P)
proof
  assume that
A1: a = a9 and
A2: b = b9 and
A3: c = c9;
  hereby
    assume
A4: a9,b9,c9 are_collinear;
A5: now
      set X = Line(a9,b9);
      assume a9<>b9;
      then reconsider P9= X as LINE of CPS by COLLSP:def 7;
      reconsider P = P9 as LINE of IncProjSp_of(CPS) by Th1;
      a9 in X by COLLSP:10;
      then
A6:   a on P by A1,Th5;
      b9 in X by COLLSP:10;
      then
A7:   b on P by A2,Th5;
      c9 in X by A4,COLLSP:11;
      then c on P by A3,Th5;
      hence ex P st a on P & b on P & c on P by A6,A7;
    end;
    now
      assume
A8:   a9=b9;
      ex P st b on P & c on P by Th9;
      hence ex P st a on P & b on P & c on P by A1,A2,A8;
    end;
    hence ex P st a on P & b on P & c on P by A5;
  end;
  given P such that
A9: a on P & b on P and
A10: c on P;
  reconsider P9=P as LINE of CPS by Th1;
A11: c9 in P9 by A3,A10,Th5;
  a9 in P9 & b9 in P9 by A1,A2,A9,Th5;
  hence thesis by A11,COLLSP:16;
end;
