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

theorem
  subset-closed_closure_of X c= X implies X is subset-closed
  proof
  set f=subset-closed_closure_of X;
  assume A1: f c=X;
  let x,y;
  assume x in X & y c=x;
  then y in f by Th2;
  hence y in X by A1;
 end;
