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:def 1;
A2: dom(f1<+>h) = dom f1 /\ dom h by Def41;
  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,Def41
    .= f.x + g.x + h.x by A5,Def41
    .= f.x + (g.x + h.x) by Th12
    .= f.x + ((g+h).x) by A6,VALUED_1:def 1
    .= (f<+>(g+h)).x by A3,A4,Def41;
end;
