theorem Th1:
  for f9 being Function st rng f c= dom f9 holds f9*f is VColoring of G
proof
  let f9 be Function;
  assume rng f c= dom f9;
  then dom(f9*f) = dom f by RELAT_1:27
    .= the_Vertices_of G by PARTFUN1:def 2;
  hence thesis by PARTFUN1:def 2;
end;
