reserve T   for TopSpace,
        A,B for Subset of T;
reserve NT,NTX,NTY for NTopSpace,
        A,B        for Subset of NT,
        O          for open Subset of NT,
        a          for Point of NT,
        XA         for Subset of NTX,
        YB         for Subset of NTY,
        x          for Point of NTX,
        y          for Point of NTY,
        f          for Function of NTX,NTY,
        fc         for continuous Function of NTX,NTY;
reserve NT for T_2 NTopSpace;
reserve T   for non empty TopSpace,
        A,B for Subset of T,
        F   for closed Subset of T,
        O   for open Subset of T;
reserve T   for non empty strict TopSpace,
        A,B for Subset of T,
        x   for Point of T;
reserve A for Subset of FMT_R^1,
        x for Point of FMT_R^1,
        y for Point of RealSpace,
        z for Point of TopSpaceMetr RealSpace,
        r for Real;

theorem
  for SF being Subset-Family of FMT_R^1
  st x = z & SF = Balls(z)
  holds <. SF .] = U_FMT x
  proof
    let SF be Subset-Family of FMT_R^1;
    assume that
A1: x = z and
A2: SF = Balls(z);
    consider zy be Point of RealSpace such that
A3: z = zy and
A4: Balls(z) = { Ball(zy,1/n) where n is Nat: n <> 0} by FRECHET:def 1;
    now
      now
        let a be object;
        assume a in {x where x is Subset of FMT_R^1:
          ex b be Element of SF st b c= x};
        then consider y be Subset of FMT_R^1 such that
A5:     a = y and
A6:     ex b be Element of SF st b c= y;
        consider b be Element of SF such that
A7:     b c= y by A6;
        b in Balls(z) by A2;
        then consider n be Nat such that
A8:     b = Ball(zy,1/n) and
A9:     n <> 0 by A4;
        reconsider r = 1 / n as Real;
        now
          now
            0 <= n & n <> 0 by A9,NAT_1:2;
            hence 0 < n by XXREAL_0:1;
            thus 0 < 1;
          end;
          hence 0 / n < 1 / n by XREAL_1:74;
          thus 0 / n = 0;
        end;
        then b is a_neighborhood of x by A1,A8,A3,Th61;
        then b in U_FMT x by FINTOPO7:def 5;
        hence a in U_FMT x by A5,A7,CARD_FIL:def 1;
      end;
      hence {x where x is Subset of FMT_R^1:
        ex b be Element of SF st b c= x} c= U_FMT x;
      now
        let o be object;
        assume
A10:    o in U_FMT x;
        then reconsider o9 = o as Subset of FMT_R^1;
        reconsider Io9 = Int o9 as open a_neighborhood of x
          by A10,FINTOPO7:def 5,Th14;
        Io9 in U_FMT x by FINTOPO7:def 5;
        then consider r be Real such that
A11:    r > 0 and
A12:    ]. x - r, x + r .[ c= Io9 by FINTOPO7:def 3,Th55;
        consider n be Nat such that
A13:    1 / n < r and
A14:    n > 0 by A11,FRECHET:36;
A15:    ]. x - 1/n , x + 1/n .[ c= ]. x - r , x + r .[ by A13,INTEGRA6:2;
        reconsider n1 = 1 / n as Real;
        ]. x - 1 / n , x + 1 / n .[ = Ball(zy,1 / n) by A1,A3,FRECHET:7;
        then ]. x - 1/n , x + 1/n .[ in { Ball(zy,1/n) where
          n is Nat: n <> 0} by A14;
        then reconsider b = ]. x - 1/n , x + 1/n .[ as Element of SF by A2,A4;
        b c= Io9 & Io9 c= o9 by A12,A15,Lm15;
        then b c= o9;
        hence o in {x where x is Subset of FMT_R^1:
          ex b be Element of SF st b c= x};
      end;
      hence U_FMT x c= {x where x is Subset of FMT_R^1:
        ex b be Element of SF st b c= x};
    end;
    hence thesis by CARDFIL2:14;
  end;
