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 Th14:
  (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;
