
theorem  :: Corollary 1
  for A being non empty finite set,
      L being Function of bool A, bool A st
    L.A = A &
    (for X, Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y) holds
  (for X being Subset of A holds L.X c= (L.(X`))`)
    iff
  L.{} = {}
  proof
    let A be non empty finite set;
    let L be Function of bool A,bool A;
    assume that
A1: L.A = A and
A2: for X, Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y;
    thus (for X being Subset of A holds L.X c= (L.(X`))` ) implies L.{} = {}
    proof
      assume for X being Subset of A holds L.X c= (L.(X`))`; then
      consider R being non empty serial finite RelStr such that
A3:   the carrier of R = A & L = LAp R by A1,Th33,A2;
      L.{} = LAp {}R by Def10,A3;
      hence thesis;
    end;
    assume L.{} = {}; then
    consider R being non empty serial RelStr such that
A4: the carrier of R = A & L = LAp R by A1,Th31,A2;
    let X be Subset of A;
    reconsider Xa = X as Subset of R by A4;
    set U = Flip L;
A5: U = UAp R by A4,Th28;
    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;
