 reserve X,a,b,c,x,y,z,t for set;
 reserve R for Relation;

theorem ThProposition9U: :: Proposition 9 H
  for A being non empty finite set,
      U being Function of bool A, bool A st
  U.{} = {} &
  (for X being Subset of A holds U.(U.X) c= U.X) &
  (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y) holds
  ex R being non empty finite transitive 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
A0: U.{} = {} &
    (for X being Subset of A holds U.(U.X) c= U.X) &
    (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y);
    set L = Flip U;
a1: L.A = A by ROUGHS_2:18,A0;
a2: for X being Subset of A holds L.X c= L.(L.X) by ROUGHS_2:24,A0;
    for X,Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y
      by A0,ROUGHS_2:21; then
    consider R being non empty finite transitive RelStr such that
W1: the carrier of R = A & L = LAp R by a1,a2,ThProposition9;
    U = Flip L by ROUGHS_2:23; then
    U = UAp R by W1,ROUGHS_2:28;
    hence thesis by W1;
  end;
