reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem
  (f-g)^2 = (g-f)^2
  proof
A1: dom(f-g) = dom f /\ dom g by VALUED_1:12;
A2: dom(g-f) = dom g /\ dom f by VALUED_1:12;
A3: dom((f-g)^2) = dom(f-g) by VALUED_1:11;
    hence dom((f-g)^2) = dom((g-f)^2) by A1,A2,VALUED_1:11;
    let x be object;
    assume
A4: x in dom((f-g)^2);
    then
A5: (f-g).x = f.x - g.x by A3,VALUED_1:13;
A6: (g-f).x = g.x - f.x by A1,A3,A4,A2,VALUED_1:13;
    thus ((f-g)^2).x = (f-g).x * (f-g).x by VALUED_1:5
    .= (g-f).x * (g-f).x by A5,A6
    .= ((g-f)^2).x by VALUED_1:5;
  end;
