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 (#) (c1 / c2)
proof
  dom(g(#)c1) = dom g & dom(g(#)c1(/)c2) = dom(g(#)c1) by VALUED_1:def 5;
  hence dom(g(#)c1(/)c2) = dom(g(#)(c1/c2)) by 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*(c1/c2)
    .= (g(#)(c1/c2)).x by VALUED_1:6;
end;
