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 Th20:
  g /" h (#) k = g (#) k /" h
proof
A1: dom(g /" h (#) k) = dom(g /" h) /\ dom k & dom(g (#) k /" h) = dom(g (#)
  k) /\ dom h by VALUED_1:16,def 4;
  dom(g /" h) = dom g /\ dom h & dom(g (#) k) = dom g /\ dom k by VALUED_1:16
,def 4;
  hence dom(g /" h (#) k) = dom(g (#) k /" h) 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_1:5
    .= g.x / h.x * k.x by VALUED_1:17
    .= g.x * k.x / h.x
    .= (g(#)k).x / h.x by VALUED_1:5
    .= (g (#) k /" h).x by VALUED_1:17;
end;
