reserve x,y, X,Y,Z for set,
        D for non empty set,
        n,k for Nat,
        i,i1,i2 for Integer;

theorem Th4:
  subset-closed_closure_of {X} = bool X
 proof
  set f=subset-closed_closure_of{X};
  A1: X in {X} by TARSKI:def 1;
  hereby let x be object;
    reconsider xx=x as set by TARSKI:1;
   assume x in f;
   then consider y such that
    A2: xx c=y and
    A3: y in {X} by Th2;
   y=X by A3,TARSKI:def 1;
   hence x in bool X by A2;
  end;
  let x be object;
  assume x in bool X;
  hence thesis by A1,Th2;
 end;
