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

theorem Th2H: :: Theorem 2 (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)`) = (U.X)`) &
    (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y) holds
  ex R being alliance finite non empty RelStr st
    the carrier of R = A & U = UAp R
  proof
    let A be non empty finite set,
        U be Function of bool A, bool A;
    assume
A1: U.{} = {} &
    (for X being Subset of A holds U.((U.X)`) = (U.X)`) &
    (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y); then
    for X being Subset of A holds U.((U.X)`) c= (U.X)`; then
    consider R being negative_alliance finite non empty RelStr such that
A2: the carrier of R = A & U = UAp R by Prop14,A1;
    for X being Subset of A holds (U.X)` c= U.((U.X)`) by A1; then
    consider S being positive_alliance finite non empty RelStr such that
A3: the carrier of S = A & U = UAp S by Prop11,A1;
A4: the RelStr of S = the RelStr of R by A2,A3,Corr3A;
    set RR = the RelStr of R;
A5: RR is positive_alliance by A4,PosReg;
    RR is negative_alliance by NegReg; then
    reconsider RR as alliance finite non empty RelStr by A5;
    UAp RR = UAp R by The5;
    hence thesis by A2;
  end;
