
theorem
  for T being TopSpace, F being Subset-Family of T st F = {} holds Fr F = {}
proof
  let T be TopSpace, F be Subset-Family of T;
  assume
A1: F = {};
  assume Fr F <> {};
  then consider x being object such that
A2: x in Fr F by XBOOLE_0:def 1;
  ex B being Subset of T st x = Fr B & B in F by A2,Def1;
  hence thesis by A1;
end;
