reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  x for set;

theorem
  F = {} implies Der F = {}
proof
  assume
A1: F = {};
  assume Der F <> {};
  then consider x being object such that
A2: x in Der F by XBOOLE_0:def 1;
  ex B being Subset of T st x = Der B & B in F by A2,Def6;
  hence thesis by A1;
end;
