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

theorem Th2:
  x in subset-closed_closure_of X iff ex y st x c= y & y in X
 proof
  set B={bool x1 where x1 is Element of X:x1 in X};
  consider F be subset-closed set such that
   A1: F=union B and
   A2: X c=F & for Y st X c=Y & Y is subset-closed holds F c=Y by Lm1;
  A3: F=subset-closed_closure_of X by A2,Def1;
  hereby assume x in subset-closed_closure_of X;
   then consider y such that
    A4: x in y and
    A5: y in B by A1,A3,TARSKI:def 4;
   consider x1 be Element of X such that
    A6: bool x1=y & x1 in X by A5;
   reconsider y=x1 as set;
   take y;
   thus x c=y & y in X by A4,A6;
  end;
  given y such that
   A7: x c=y and
   A8: y in X;
  bool y in B by A8;
  hence thesis by A1,A3,A7,TARSKI:def 4;
 end;
