
theorem
  for X being set,
      f being Function of bool X, bool X st
    for A, B being Subset of X holds f.(A \/ B) = f.A \/ f.B holds
  for A, B being Subset of X holds
  (Flip f).(A /\ B) = (Flip f).A /\ (Flip f).B
  proof
    let X be set,
        f be Function of bool X, bool X;
    assume
A1: for A, B being Subset of X holds f.(A \/ B) = f.A \/ f.B;
    let A, B be Subset of X;
    set g = Flip f;
    g.(A /\ B) = (f.(A /\ B)`)` by Def14
              .= (f.(A` \/ B`))` by XBOOLE_1:54
              .= (f.(A`) \/ f.(B`))` by A1
              .= (f.(A`))` /\ (f.(B`))` by XBOOLE_1:53
              .= g.A /\ (f.(B`))` by Def14
              .= g.A /\ g.B by Def14;
    hence thesis;
  end;
