reserve r for Real,
  X for set,
  f, g, h for real-valued Function;

theorem
  f c= g implies f-h c= g-h
proof
A1: dom(f-h) = dom f /\ dom h by VALUED_1:12;
A2: dom(g-h) = dom g /\ dom h by VALUED_1:12;
  assume
A3: f c= g;
  then dom f c= dom g by GRFUNC_1:2;
  then
A4: dom(f-h) c= dom(g-h) by A1,A2,XBOOLE_1:27;
  now
    let x be object;
    assume
A5: x in dom(f-h);
    then
A6: x in dom f by A1,XBOOLE_0:def 4;
    thus (f-h).x = f.x - h.x by A5,VALUED_1:13
      .= g.x - h.x by A3,A6,GRFUNC_1:2
      .= (g-h).x by A4,A5,VALUED_1:13;
  end;
  hence thesis by A4,GRFUNC_1:2;
end;
