
theorem
  for N being RealNormSpace,
      s being sequence of the carrier of TopSpaceMetr MetricSpaceNorm N,
      x being Point of TopSpaceMetr MetricSpaceNorm N holds
    x in lim_f s
      iff
    (for n being positive Nat ex i be Nat st for j being Nat st i <= j holds
     ||.#x- #(s.j) .|| < 1/n)
  proof
    let N be RealNormSpace,
    s be sequence of the carrier of TopSpaceMetr MetricSpaceNorm(N),
    x be Point of TopSpaceMetr MetricSpaceNorm N;
    reconsider x1=x as Point of TopSpaceMetr MetricSpaceNorm N;
    consider y0 be Point of MetricSpaceNorm N such that
A1: y0=x1 and
A2: Balls x1={Ball(y0,1/n) where n is Nat:n <> 0} by FRECHET:def 1;
A3: x in lim_f s implies
    (for n be positive Nat ex i be Nat st
    for j be Nat st i <= j holds ||.#x- #(s.j) .|| < 1/n)
    proof
      assume
A4:   x in lim_f s;
      now
        let n be positive Nat;
        Ball(y0,1/n) in Balls x1 by A2;
        then consider i0 be Nat such that
A5:     for j be Nat st i0 <= j holds s.j in Ball(y0,1/n) by A4,Th6;
A6:     now
          let j be Nat;
          assume
A7:       i0 <= j;
          consider y1 be Point of N such that
A8:       y0=y1 and
A9:       Ball (y0,1/n)={q where q is Point of N : ||.y1 - q.|| < 1/n}
          by NORMSP_2:2;
          s.j in {q where q is Point of N : ||.y1 - q.|| < 1/n} by A7,A5,A9;
          then consider q0 be Point of N such that
A10:       s.j=q0 and
A11:       ||.y1  - q0.|| < 1/n;
          thus ||.#x - #(s.j).|| < 1/n by A1,A8,A10,A11;
        end;
        take i0;
        thus for j be Nat st i0<=j holds ||.#x - #(s.j).|| < 1/n by A6;
      end;
      hence thesis;
    end;
    (for n be positive Nat ex i be Nat st for j be Nat st i<=j holds
    ||.#x - #(s.j).|| < 1/n) implies x in lim_f s
    proof
      assume
A12:  for n be positive Nat ex i be Nat st
      for j be Nat st i<=j holds
      ||.#x - #(s.j).|| < 1/n;
      for b be Element of Balls(x) ex i be Nat st
      for j be Nat st i <=j holds s.j in b
      proof
        let b be Element of Balls(x);
        b in {Ball(y0,1/n) where n is Nat:n <> 0} by A2;
        then consider n0 be Nat such that
A13:     b=Ball(y0,1/n0) and
A14:     n0 <> 0;
        consider i0 be Nat such that
A15:     for j be Nat st i0<=j holds ||.#x - #(s.j).|| < 1/n0 by A12,A14;
        take i0;
        for j be Nat st i0 <= j holds s.j in b
        proof
          let j be Nat;
          assume i0<=j; then
A16:      ||.#x1 - #(s.j).||< 1/n0 by A15;
          consider y1 be Point of N such that
A17:      y0=y1 and
A18:      Ball (y0,1/n0)={q where q is Point of N : ||.y1 - q.|| < 1/n0}
          by NORMSP_2:2;
          thus s.j in b by A1,A13,A16,A17,A18;
        end;
        hence thesis;
      end;
      hence thesis by Th6;
    end;
    hence thesis by A3;
  end;
