
theorem  :: Corollary 2
  for A being non empty finite set,
      U being Function of bool A, bool A st
    U.{} = {} &
    (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y) holds
  (for X being Subset of A holds (U.(X`))` c= U.X)
    iff
  U.A = A
  proof
    let A be non empty finite set;
    let U be Function of bool A,bool A;
    assume that
A1: U.{} = {} and
A2: for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y;
    thus (for X being Subset of A holds (U.(X`))` c= U.X) implies U.A = A
    proof
      assume for X being Subset of A holds (U.(X`))` c= U.X; then
      consider R being non empty serial RelStr such that
A3:   the carrier of R = A & U = UAp R by Th34,A1,A2;
      (UAp R).[#]R = UAp [#]R by Def11;
      hence thesis by A3;
    end;
    assume U.A = A; then
    consider R being non empty finite serial RelStr such that
A4: the carrier of R = A & U = UAp R by A1,Th32,A2;
    let X be Subset of A;
    reconsider Xa = X as Subset of R by A4;
    set L = Flip U;
A5: L = LAp R by A4,Th27;
    LAp Xa c= UAp Xa by Th17; then
    LAp Xa c= (UAp R).X by Def11; then
    (LAp R).X c= (UAp R).X by Def10;
    hence thesis by Def14,A4,A5;
  end;
