theorem
  rng f c= rng g implies
  for x being object st x in dom f
   ex y being object st y in dom g & f.x = g. y
proof
  assume that
A1: rng f c= rng g;
  let x be object;
  assume x in dom f;
  then f.x in rng f by Def3;
  then
A2: f.x in rng g by A1;
  ex y being object st y in dom g & f.x = g.y by Def3,A2;
 hence thesis;
end;
