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_1:16,def 5;
  hence dom((g/"h)(/)c) = dom(g/"(h(#)c)) by A1,VALUED_1:16;
  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_1:17
    .= g.x/(h.x*c) by XCMPLX_1:78
    .= g.x / (h(#)c).x by VALUED_1:6
    .= (g/"(h(#)c)).x by VALUED_1:17;
end;
