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 Th61:
  x = y & r > 0 implies Ball(y,r) is a_neighborhood of x
  proof
    assume that
A1: x = y and
A2: r > 0;
    reconsider S = ]. x - r , x + r .[ as Subset of FMT_R^1
      by TOPMETR:17,FINTOPO7:def 15;
    S is a_neighborhood of x by A2,Th58;
    hence thesis by A1,FRECHET:7;
  end;
