
theorem Th30:  :: Proposition 1 1L 4L
  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
  ex R being non empty finite RelStr st
  the carrier of R = A & L = LAp R
  proof
    let A be non empty finite set,
        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;
    set U = Flip L;
A3: U.{} = {} by Th19,A1;
A4: for X, Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y by Th22,A2;
    consider R being non empty finite RelStr such that
A5: the carrier of R = A & U = UAp R by Th29,A3,A4;
    take R;
    L = Flip UAp R by Th23,A5;
    hence thesis by A5,Th27;
  end;
