reserve X for non empty set;

theorem
  for ET being FMT_TopSpace, cF being Filter of the carrier of ET,
  cS being non empty Subset of cF holds for A being non empty Subset of ET
  st cF = Neighborhood A & cS = OpenNeighborhoods A holds cS is filter_basis
  proof
    let ET be FMT_TopSpace,
    cF be Filter of the carrier of ET,
    cS be non empty Subset of cF;
    let A be non empty Subset of ET
    such that
A1: cF = Neighborhood A and
A2: cS = OpenNeighborhoods A;
    for f be Element of cF holds ex b be Element of cS st b c= f
    proof
      let f be Element of cF;
      f in the set of all N where N is a_neighborhood of A by A1;
      then consider N be a_neighborhood of A such that
A3:   f = N;
      consider O be open Subset of ET such that
A4:   A c= O & O c= N by Th11;
      O is open;
      then for x be Element of ET st x in A holds O in U_FMT x by A4;
      then O is open a_neighborhood of A by Def6;
      then O in the set of all N where N is open a_neighborhood of A;
      hence thesis by A2,A3,A4;
    end;
    hence thesis;
  end;
