
theorem Th31:  :: Proposition 2 1L 4L 2L
  for A being non empty finite set,
      L being Function of bool A, bool A st
    L.A = A &
    L.{} = {} &
    (for X, Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y) holds
  ex R being non empty serial RelStr st
  the carrier of R = A & L = LAp R
  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: L.{} = {} and
A3: for X, Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y;
    consider R being non empty finite RelStr such that
A4: the carrier of R = A & L = LAp R by Th30,A3,A1;
    for x being object st x in the carrier of R
      ex y being object st
        y in the carrier of R & [x,y] in the InternalRel of R
    proof
      let x be object;
      assume
A5:   x in the carrier of R;
A6:   (LAp R).{} = LAp {}R by Def10
                .= { y where y is Element of R :
        Class (the InternalRel of R, y) c= {} };
      for y being Element of R holds Class (the InternalRel of R, y) <> {}
      proof
        let y be Element of R;
        assume Class (the InternalRel of R, y) = {}; then
        y in { y where y is Element of R :
          Class (the InternalRel of R, y) c= {} };
        hence contradiction by A6,A4,A2;
      end; then
      Class (the InternalRel of R, x) <> {} by A5; then
      consider t be object such that
A7:   t in Im(the InternalRel of R,x) by XBOOLE_0:def 1;
A8:   [x,t] in the InternalRel of R by A7,RELAT_1:169; then
      t in rng the InternalRel of R by A5,RELSET_1:25;
      hence thesis by A8;
    end; then
    R is serial by Def1;
    hence thesis by A4;
  end;
