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

theorem Th3:
  X c=Y & Y is subset-closed implies subset-closed_closure_of X c= Y
 proof
  assume A1: X c=Y & Y is subset-closed;
  let x be object;
    reconsider xx=x as set by TARSKI:1;
  assume x in subset-closed_closure_of X;
  then ex y st xx c=y & y in X by Th2;
  hence thesis by A1;
 end;
