reserve X for non empty set;

theorem Th7:
  for ET being U_FMT_filter non empty strict FMT_Space_Str,
  A being Subset of ET holds for NA being a_neighborhood of A,
  B being Subset of ET st NA c= B holds B is a_neighborhood of A
  proof
    let ET be U_FMT_filter non empty strict FMT_Space_Str,A be Subset of ET;
    let NA be a_neighborhood of A,B be Subset of ET;
    assume
A1: NA c= B;
    for x be Element of ET st x in A holds B in U_FMT x
    proof
      let x be Element of ET;
      assume x in A; then
A2:   NA in U_FMT x by Def6;
      U_FMT x is Filter of the carrier of ET by Def2;
      hence thesis by A1,A2,CARD_FIL:def 1;
    end;
    hence thesis by Def6;
  end;
