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
  gen_filter(gen_R^1) = FMT_R^1
  proof
    now
      the carrier of FMT_R^1 = REAL by TOPMETR:17,FINTOPO7:def 15;
      then dom the BNbd of FMT_R^1 = REAL by FUNCT_2:def 1;
      hence dom <. gen_R^1 .] = dom the BNbd of FMT_R^1 by Th64,FUNCT_2:def 1;
      hereby
        let x be object;
        assume x in dom <. gen_R^1 .];
        then reconsider y = x as Element of the carrier of gen_R^1;
        reconsider z = y as Element of FMT_R^1;
A1:     (the BNbd of FMT_R^1). z = U_FMT z by FINTOPO2:def 6;
        now
          now
            let o be object;
            assume
A2:         o in U_FMT z;
            then reconsider S = o as Subset of FMT_R^1;
            S is a_neighborhood of z by A2,FINTOPO7:def 5;
            then consider O be open Subset of FMT_R^1 such that
A3:         z in O and
A4:         O c= S by FINTOPO7:15;
A5:         NTop2Top FMT_R^1 = R^1 by FINTOPO7:24;
            O is open Subset of R^1 by A5,Lm9;
            then consider r be Real such that
A6:         r > 0 and
A7:         ]. z - r, z + r .[ c= O by A3,FRECHET:8;
            consider n be Nat such that
A8:         1 / n < r and
A9:         n > 0 by A6,FRECHET:36;
A10:        ]. z - 1/n , z + 1/n .[ c= ]. z - r , z + r .[
              by A8,INTEGRA6:2;
            reconsider z9 = z as Point of TopSpaceMetr(RealSpace)
              by TOPMETR:def 6,FINTOPO7:def 15;
            consider zy be Point of RealSpace such that
A11:        z9 = zy and
A12:        Balls(z9) = { Ball(zy,1/n) where n is Nat: n <> 0}
              by FRECHET:def 1;
A13:        ]. z - 1 / n , z + 1 / n .[ = Ball(zy,1 / n) by A11,FRECHET:7;
            consider py be Point of TopSpaceMetr(RealSpace) such that
A14:        py = y and
A15:        U_FMT y = Balls(py) by Th65;
            ]. z - 1/n , z + 1/n .[ in Balls(py) by A13,A9,A12,A14;
            then reconsider b = ]. z - 1/n , z + 1/n .[ as
              Element of U_FMT y by A15;
            b c= S by A10,A7,A4;
            hence o in <. (U_FMT y) .] by CARDFIL2:def 8;
          end;
          hence U_FMT z c= <. (U_FMT y) .];
          now
            let o be object;
            assume
A16:        o in <. (U_FMT y) .];
            then reconsider O = o as Subset of gen_R^1;
            consider b be Element of U_FMT y such that
A17:        b c= O by A16,CARDFIL2:def 8;
            reconsider z9 = z as Point of TopSpaceMetr(RealSpace)
              by FINTOPO7:def 15,TOPMETR:def 6;
            consider zy be Point of RealSpace such that
A18:        z9 = zy and
A19:        Balls(z9) = { Ball(zy,1/n) where n is Nat: n <> 0}
              by FRECHET:def 1;
            consider py be Point of TopSpaceMetr(RealSpace) such that
A20:        py = y and
A21:        U_FMT y = Balls(py) by Th65;
            b in Balls(z9) by A21,A20;
            then consider n be Nat such that
A22:        b = Ball(zy,1/n) and
A23:        n <> 0 by A19;
A24:        ]. z - 1 / n , z + 1 / n .[ = Ball(zy,1 / n)
              by A18,FRECHET:7;
A25:        n > 0 by NAT_1:3,A23;
            0 / n < 1 / n by A25,XREAL_1:74;
            then Ball(zy,1/n) is a_neighborhood of z by A24,Th58;
            then Ball(zy,1/n) in U_FMT z by FINTOPO7:def 5;
            hence o in U_FMT z by A22,A17,CARD_FIL:def 1;
          end;
          hence <. (U_FMT y) .] c= U_FMT z;
        end;
        hence <. gen_R^1 .] . x = (the BNbd of FMT_R^1). x
          by FINTOPO7:def 9,A1;
      end;
    end;
    then FMT_Space_Str(# the carrier of gen_R^1,<. gen_R^1 .] #) = FMT_R^1
      by FUNCT_1:2;
    hence thesis by FINTOPO7:def 10;
  end;
