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 Th13:
  (g + c1) - c2 = g + (c1 - c2)
proof
A1: dom((g+c1)-c2) = dom(g+c1) by VALUED_1:def 2;
A2: dom(g+c1) = dom g by VALUED_1:def 2;
  hence dom((g+c1)-c2) = dom(g+(c1-c2)) by A1,VALUED_1:def 2;
  let x be object;
A3: dom(g+(c1-c2)) = dom(g) by VALUED_1:def 2;
  assume
A4: x in dom((g+c1)-c2);
  hence ((g+c1)-c2).x = (g+c1).x-c2 by VALUED_1:def 2
    .= g.x+c1-c2 by A1,A4,VALUED_1:def 2
    .= g.x+(c1-c2)
    .= (g+(c1-c2)).x by A1,A2,A3,A4,VALUED_1:def 2;
end;
