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 Th25:
  - (g (#) h) = (-g) (#) h
proof
A1: dom -(g(#)h) = dom(g(#)h) by VALUED_1:8;
  dom(g(#)h) = dom g /\ dom h & dom((-g)(#)h) = dom(-g) /\ dom h by
VALUED_1:def 4;
  hence dom -(g(#)h) = dom((-g)(#)h) by A1,VALUED_1:8;
  let x be object;
  assume x in dom -(g(#)h);
  thus (-(g(#)h)).x = -(g(#)h).x by VALUED_1:8
    .= -(g.x*h.x) by VALUED_1:5
    .= (-g.x)*h.x
    .= (-g).x*h.x by VALUED_1:8
    .= ((-g)(#)h).x by VALUED_1:5;
end;
