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 Th18:
  g - h = - (h - g)
proof
A1: dom -(h-g) = dom(h-g) by VALUED_1:8;
  dom(g-h) = dom g /\ dom h by VALUED_1:12;
  hence
A2: dom(g-h) = dom -(h-g) by A1,VALUED_1:12;
  let x be object;
  assume
A3: x in dom(g-h);
  hence (g-h).x = g.x-h.x by VALUED_1:13
    .= -(h.x-g.x)
    .= -(h-g).x by A1,A2,A3,VALUED_1:13
    .= (-(h-g)).x by VALUED_1:8;
end;
