theorem
  union bool A = A
proof
  now
    let x;
    thus x in union bool A implies x in A
    proof
      assume x in union bool A;
      then consider X such that
A1:   x in X and
A2:   X in bool A by TARSKI:def 4;
      X c= A by A2,Def1;
      hence thesis by A1;
    end;
    assume x in A;
    then { x } c= A by Lm1;
    then
A3: { x } in bool A by Def1;
    x in { x } by TARSKI:def 1;
    hence x in union bool A by A3,TARSKI:def 4;
  end;
  hence thesis by TARSKI:2;
end;
