theorem Th24:
  f c= g implies Union f c= Union g
proof
  assume
A1: f c= g;
  then
A2: dom f c= dom g by GRFUNC_1:2;
  let x be object;
  assume x in Union f;
  then consider X such that
A3: x in X and
A4: X in rng f by TARSKI:def 4;
  consider y being object such that
A5: y in dom f and
A6: X = f.y by A4,FUNCT_1:def 3;
  f.y = g.y by A1,A5,GRFUNC_1:2;
  then X in rng g by A2,A5,A6,FUNCT_1:def 3;
  hence thesis by A3,TARSKI:def 4;
end;
