reserve x, y for object, X, X1, X2 for set;
reserve Y, Y1, Y2 for complex-functions-membered set,
  c, c1, c2 for Complex,
  f for PartFunc of X,Y,
  f1 for PartFunc of X1,Y1,
  f2 for PartFunc of X2, Y2,
  g, h, k for complex-valued Function;

theorem
  f <+> g <-> h = f <+> (g-h)
proof
  set f1 = f<+>g;
A1: dom(g-h) = dom g /\ dom h by VALUED_1:12;
A2: dom(f1<->h) = dom f1 /\ dom h by Th61;
  dom f1 = dom f /\ dom g & dom(f<+>(g-h)) = dom f /\ dom(g-h) by Def41;
  hence
A3: dom(f1<->h) = dom(f<+>(g-h)) by A2,A1,XBOOLE_1:16;
  let x be object;
  assume
A4: x in dom(f1<->h);
  then
A5: x in dom f1 by A2,XBOOLE_0:def 4;
A6: x in dom(g-h) by A3,A4,XBOOLE_0:def 4;
  thus (f1<->h).x = f1.x - h.x by A4,Th62
    .= f.x + g.x - h.x by A5,Def41
    .= f.x + (g.x - h.x) by Th13
    .= f.x + ((g-h).x) by A6,VALUED_1:13
    .= (f<+>(g-h)).x by A3,A4,Def41;
end;
