
theorem
  for T being TopSpace, F being Subset-Family of T, A being Subset of T
  st F = { A } holds Fr F = { Fr A }
proof
  let T be TopSpace, F be Subset-Family of T, A be Subset of T;
  assume
A1: F = { A };
  thus Fr F c= { Fr A }
  proof
    let x be object;
    assume
A2: x in Fr F;
    then reconsider B = x as Subset of T;
    consider C being Subset of T such that
A3: B = Fr C and
A4: C in F by A2,Def1;
    C = A by A1,A4,TARSKI:def 1;
    hence thesis by A3,TARSKI:def 1;
  end;
  let x be object;
  assume x in { Fr A };
  then
A5: x = Fr A by TARSKI:def 1;
  A in F by A1,TARSKI:def 1;
  hence thesis by A5,Def1;
end;
