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

theorem :: Proposition 12
  for A being non empty finite set,
      L being Function of bool A, bool A st
    L.A = A &
    (for X being Subset of A holds L.((L.X)`) c= (L.X)`) &
    (for X,Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y) holds
  ex R being positive_alliance finite non empty 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
A1: L.A = A &
    (for X being Subset of A holds L.((L.X)`) c= (L.X)`) &
    for X,Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y;
    set U = Flip L;
A2: for X being Subset of A holds (U.X)` c= U.((U.X)`) by A1,Conv4;
A4: U.{} = {} by A1,ROUGHS_2:19;
    (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y)
      by ROUGHS_2:22,A1; then
    consider R being positive_alliance finite non empty RelStr such that
A3: the carrier of R = A & U = UAp R by Prop11,A2,A4;
    L = Flip UAp R by A3,ROUGHS_2:23; then
    L = LAp R by ROUGHS_2:27;
    hence thesis by A3;
  end;
