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

theorem :: Theorem 2 (L)
  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.X)` = L.((L.X)`)) &
    (for X,Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y) holds
  ex R being 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
A0: L.A = A &
    (for X being Subset of A holds (L.X)` = L.((L.X)`)) &
    (for X,Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y);
    set U = Flip L;
A2: U.{} = {} by A0,ROUGHS_2:19;
A3: for X being Subset of A holds U.((U.X)`) = (U.X)`
    proof
      let X be Subset of A;
      for X being Subset of A holds (L.X)` c= L.((L.X)`) by A0; then
Z1:   U.((U.X)`) c= (U.X)` by Conv2;
Z2:   L = Flip U by ROUGHS_2:23;
      for X being Subset of A holds L.((L.X)`) c= (L.X)` by A0; then
      (U.X)` c= U.((U.X)`) by Conv3,Z2;
      hence thesis by Z1,XBOOLE_0:def 10;
    end;
    for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y
      by A0,ROUGHS_2:22; then
    consider R being alliance finite non empty RelStr such that
A1: the carrier of R = A & U = UAp R by A2,A3,Th2H;
    Flip U = L by ROUGHS_2:23; then
    L = LAp R by A1,ROUGHS_2:27;
    hence thesis by A1;
  end;
