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 (#) (h(/)c)
proof
A1: dom((g(#)h)(/)c) = dom(g(#)h) by VALUED_1:def 5;
  dom(g(#)h) = dom g /\ dom h & dom(h(/)c) = dom h by VALUED_1:def 4,def 5;
  hence dom((g(#)h)(/)c) = dom(g(#)(h(/)c)) by A1,VALUED_1:def 4;
  let x be object;
  assume 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 VALUED_1:5
    .= g.x*(h.x*c")
    .= g.x * (h(/)c).x by VALUED_1:6
    .= (g(#)(h(/)c)).x by VALUED_1:5;
end;
