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(/)c)^2 = f^2 (/) c^2
  proof
    thus dom((f(/)c)^2) = dom(f(/)c) by VALUED_1:11
    .= dom f by VALUED_2:28
    .= dom(f^2) by VALUED_1:11
    .= dom(f^2 (/) c^2) by VALUED_2:28;
    let x be object;
    assume x in dom((f(/)c)^2);
    thus ((f(/)c)^2).x = ((f(/)c).x)^2 by VALUED_1:11
    .= (f.x/c)^2 by VALUED_2:29
    .= (f.x)^2 / c^2 by XCMPLX_1:76
    .= (f^2).x / c^2 by VALUED_1:11
    .= (f^2 (/) c^2).x by VALUED_2:29;
  end;
