
theorem Th32:  :: Proposition 2 1H 4H 2H
  for A being non empty finite set,
      U being Function of bool A, bool A st
    U.A = A &
    U.{} = {} &
    (for X, Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y) holds
  ex R being non empty finite serial RelStr st
  the carrier of R = A & U = UAp R
  proof
    let A be non empty finite set;
    let U be Function of bool A,bool A;
    assume that
A1: U.A = A and
A2: U.{} = {} and
A3: for X, Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y;
    consider R being non empty finite RelStr such that
A4: the carrier of R = A & U = UAp R by Th29,A2,A3;
    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;
      reconsider Z = [#]A as Subset of R by A4;
      UAp Z = Z by A4,A1,Def11; then
      consider t being Element of R such that
A6:   t = x & Class (the InternalRel of R,t) meets [#]A by A5,A4;
      Class (the InternalRel of R,t) <> {} by A6; then
      consider s being object such that
A7:   s in Class (the InternalRel of R,t) by XBOOLE_0:def 1;
      [t,s] in the InternalRel of R by A7,RELAT_1:169; then
      [x,s] in the InternalRel of R & s in rng the InternalRel of R
        by XTUPLE_0:def 13,A6;
      hence thesis;
    end; then
    R is serial by Def1;
    hence thesis by A4;
  end;
