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 (/) c1 (#) c2 = g (#) c2 (/) c1
proof
A1: dom(g(/)c1(#)c2) = dom(g(/)c1) by VALUED_1:def 5;
  dom(g(/)c1) = dom g & dom(g(#)c2) = dom g by VALUED_1:def 5;
  hence dom(g(/)c1(#)c2) = dom(g(#)c2(/)c1) by A1,VALUED_1:def 5;
  let x be object;
  assume x in dom(g(/)c1(#)c2);
  thus (g(/)c1(#)c2).x = (g(/)c1).x * c2 by VALUED_1:6
    .= g.x * c1" * c2 by VALUED_1:6
    .= (g.x*c2)*c1"
    .= (g(#)c2).x*c1" by VALUED_1:6
    .= (g(#)c2(/)c1).x by VALUED_1:6;
end;
