reserve G for IncProjStr;
reserve a,a1,a2,b,b1,b2,c,d,p,q,r for POINT of G;
reserve A,B,C,D,M,N,P,Q,R for LINE of G;

theorem Th5:
  a,b,c are_collinear iff ex P st a on P & b on P & c on P
proof
  (ex P st {a,b,c} on P) iff ex P st a on P & b on P & c on P
  proof
    hereby
      given P such that
A1:   {a,b,c} on P;
      take P;
      thus a on P & b on P & c on P by A1,INCSP_1:2;
    end;
    thus (ex P st a on P & b on P & c on P) implies
    ex P st {a,b,c} on P by INCSP_1:2;
  end;
  hence thesis;
end;
