theorem
  for g1,g2 being Function of M./.R, N st g1 * nat_hom R = g2 * nat_hom R
  holds g1 = g2
proof
  let g1,g2 be Function of M./.R, N;
  assume A1: g1 * nat_hom R = g2 * nat_hom R;
  set Y = rng nat_hom R;
  rng nat_hom R = the carrier of M ./. R by FUNCT_2:def 3; then
  dom g1 = Y & dom g2 = Y by FUNCT_2:def 1;
  hence g1 = g2 by A1,FUNCT_1:86;
end;
