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 Th19:
  g (#) h /" k = g (#) (h /" k)
proof
A1: dom(g (#) (h /" k)) = dom g /\ dom(h /" k) & dom(g (#) h /" k) = dom(g
  (#) h ) /\ dom k by VALUED_1:16,def 4;
  dom(g (#) h) = dom g /\ dom h & dom(h /" k) = dom h /\ dom k by VALUED_1:16
,def 4;
  hence dom(g (#) h /" k) = dom(g (#) (h /" k)) by A1,XBOOLE_1:16;
  let x be object;
  assume x in dom(g (#) h /" k);
  thus (g (#) h /" k).x = (g (#) h).x / k.x by VALUED_1:17
    .= g.x * h.x / k.x by VALUED_1:5
    .= g.x * (h.x / k.x)
    .= g.x * (h /" k).x by VALUED_1:17
    .= (g (#) (h /" k)).x by VALUED_1:5;
end;
