reserve X for non empty set;

theorem Th7bis:
  for ET being U_FMT_filter non empty strict FMT_Space_Str,
  A being non empty Subset of ET holds
  Neighborhood A is Filter of the carrier of ET
  proof
    let ET be U_FMT_filter non empty strict FMT_Space_Str,
    A be non empty Subset of ET;
    set S=the set of all F where F is a_neighborhood of A;
    S is Filter of the carrier of ET
    proof
A1:   S is non empty Subset-Family of the carrier of ET
      proof
A2:     S is non empty
        proof
          set Sq=the carrier of ET;
          Sq c= the carrier of ET;
          then reconsider Sq as Subset of the carrier of ET;
          Sq is a_neighborhood of A
          proof
            for x be Element of ET st x in A holds Sq in U_FMT x by Th6;
            hence thesis by Def6;
          end;
          then Sq in S;
          hence thesis;
        end;
        S is Subset-Family of the carrier of ET
        proof
          S c= bool the carrier of ET
          proof
            let t be object;
            assume t in S;
            then consider F0 be a_neighborhood of A such that
A3:         t=F0;
            thus t in bool the carrier of ET by A3;
          end;
          hence thesis;
        end;
        hence thesis by A2;
      end;
A4:   not {} in S
      proof
        assume {} in S;
        then consider F0 be a_neighborhood of A such that
A5:     {}=F0;
        A is non empty;
        then consider a be object such that
A6:     a in A;
        reconsider a0=a as Element of ET by A6;
A7:     {} in U_FMT a0 by A5,A6,Def6;
        U_FMT a0 is Filter of the carrier of ET by Def2;
        hence thesis by A7,CARD_FIL:def 1;
      end;
       for Y1,Y2 being Subset of ET holds
      (Y1 in S & Y2 in S implies Y1/\Y2 in S) &
      (Y1 in S & Y1 c= Y2 implies Y2 in S)
      proof
        let Y1,Y2 being Subset of ET;
        now
          assume that
A8:       Y1 in S and
A9:       Y2 in S;
          consider F1 be a_neighborhood of A such that
A10:      Y1=F1 by A8;
          consider F2 be a_neighborhood of A such that
A11:      Y2=F2 by A9;
A12:      for x be Element of ET st x in A holds Y1/\Y2 in U_FMT x
          proof
            let x be Element of ET;
            assume x in A; then
A13:        Y1 in U_FMT x & Y2 in U_FMT x by A10,A11,Def6;
            U_FMT x is Filter of the carrier of ET by Def2;
            hence thesis by A13,CARD_FIL:def 1;
          end;
          Y1/\Y2 is a_neighborhood of A by A12,Def6;
          hence Y1/\Y2 in S;
        end;
        hence Y1 in S & Y2 in S implies Y1/\Y2 in S;
        now
          assume that
A14:      Y1 in S and
A15:      Y1 c= Y2;
          consider y1 be a_neighborhood of A such that
A16:      y1=Y1 by A14;
          for x be Element of ET st x in A holds Y2 in U_FMT x
          proof
            let x be Element of ET;
            assume x in A; then
A17:        Y1 in U_FMT x by A16,Def6;
            U_FMT x is Filter of the carrier of ET by Def2;
            hence thesis by A17,A15,CARD_FIL:def 1;
          end;
          then Y2 is a_neighborhood of A by Def6;
          hence Y1 in S & Y1 c= Y2 implies Y2 in S;
        end;
        hence thesis;
      end;
      hence thesis by A1,A4,CARD_FIL:def 1;
    end;
    hence thesis;
  end;
