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

theorem :: Proposition 15
  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)` c= L.((L.X)`)) &
    (for X,Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y) holds
  ex R being negative_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 that
A1: L.A = A and
A3: for X being Subset of A holds (L.X)` c= L.((L.X)`) and
A4: (for X,Y being Subset of A holds L.(X /\ Y) = L.X /\ L.Y);
    set U = Flip L;
C1: U.{} = {} by A1,ROUGHS_2:19;
C2: for X being Subset of A holds U.((U.X)`) c= (U.X)` by A3,Conv2;
    (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y)
      by A4,ROUGHS_2:22; then
    consider R being negative_alliance finite non empty RelStr such that
A2: the carrier of R = A & U = UAp R by Prop14,C1,C2;
    Flip U = LAp R by A2,ROUGHS_2:27; then
    L = LAp R by ROUGHS_2:23;
    hence thesis by A2;
  end;
