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

theorem Th6:
  X is_finer_than Y iff
    subset-closed_closure_of X c= subset-closed_closure_of Y
 proof
  set fx=subset-closed_closure_of X;
  set fy=subset-closed_closure_of Y;
  hereby assume A1: X is_finer_than Y;
   thus fx c=fy
   proof
    let x be object;
    reconsider xx=x as set by TARSKI:1;
    assume x in fx;
    then consider y such that
     A2: xx c=y and
     A3: y in X by Th2;
    consider c be set such that
     A4: c in Y and
     A5: y c=c by A1,A3;
    xx c=c by A2,A5;
    hence thesis by A4,Th2;
   end;
  end;
  assume A6: fx c=fy;
  let x;
  assume x in X;
  then x in fx by Th2;
  then ex y st x c=y & y in Y by A6,Th2;
  hence thesis;
 end;
