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 Th23:
  c(#)-g = -(c(#)g)
proof
A1: dom(-(c(#)g)) = dom(c(#)g) by VALUED_1:8
    .= dom g by VALUED_1:def 5;
  dom(c(#)-g) = dom -g by VALUED_1:def 5
    .= dom g by VALUED_1:8;
  hence dom(c(#)-g) = dom -(c(#)g) by A1;
  let x be object;
  assume x in dom(c(#)-g);
  thus (c(#)-g).x = c*((-g).x) by VALUED_1:6
    .= c*(-g.x) by VALUED_1:8
    .= -(c*g.x)
    .= -(c(#)g).x by VALUED_1:6
    .= (-(c(#)g)).x by VALUED_1:8;
end;
