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
  (g+h) (/) c = g(/)c + h(/)c
proof
A1: dom((g+h)(/)c) = dom(g+h) by VALUED_1:def 5;
A2: dom(g+h) = dom g /\ dom h by VALUED_1:def 1;
  dom(g(/)c) = dom g & dom(h(/)c) = dom h by VALUED_1:def 5;
  hence
A3: dom((g+h)(/)c) = dom(g(/)c+h(/)c) by A1,A2,VALUED_1:def 1;
  let x be object;
  assume
A4: x in dom((g+h)(/)c);
  thus ((g+h)(/)c).x = (g+h).x * c" by VALUED_1:6
    .= (g.x+h.x)*c" by A1,A4,VALUED_1:def 1
    .= g.x*c" + h.x*c"
    .= (g(/)c).x + h.x*c" by VALUED_1:6
    .= (g(/)c).x + (h(/)c).x by VALUED_1:6
    .= (g(/)c+h(/)c).x by A3,A4,VALUED_1:def 1;
end;
