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

theorem
  Y is subset-closed iff for X st X in Y holds bool X c= Y
 proof
  thus Y is subset-closed implies for X st X in Y holds bool X c=Y;
  assume A1: for X st X in Y holds bool X c=Y;
  let x,y;
  assume that
   A2: x in Y and
   A3: y c=x;
  A4: y in bool x by A3;
  bool x c=Y by A1,A2;
  hence thesis by A4;
 end;
