
theorem Th23:
  for S1 be sequence of RealSpace, seq be Real_Sequence,
      g be Real, g1 be Element of RealSpace st S1 = seq & g = g1 holds
      (for p be Real st 0 < p
       ex n be Nat st for m be Nat st n <= m holds |.seq.m - g.| < p)
    iff
      (for p be Real st 0 < p
       ex n be Nat st for m be Nat st n <= m holds dist(S1.m,g1) < p)
  proof
    let S1 be sequence of RealSpace, seq be Real_Sequence,
        g be Real, g1 be Element of RealSpace;
    assume
    A1: S1 = seq & g = g1;
    hereby
      assume
      A2: for p be Real st 0 < p
          ex n be Nat st for m be Nat st n <= m holds |.seq.m - g.| < p;
      thus for p be Real st 0 < p
      ex n be Nat st for m be Nat st n <= m holds dist(S1.m,g1) < p
      proof
        let p be Real;
        assume 0 < p; then
        consider n be Nat such that
        A3: for m be Nat st n <= m holds |.seq.m - g.| < p by A2;
        A4: for m be Nat st n <= m holds dist(S1.m,g1) < p
        proof
          let m be Nat;
          assume C5: n <= m;
          set s = seq.m;
          set s1 = S1.m;
          dist(s1,g1) = |.s - g.| by A1,TOPMETR:11;
          hence thesis by A3,C5;
        end;
        take n;
        thus thesis by A4;
      end;
    end;
    assume
    A5: for p be Real st 0 < p
        ex n be Nat st for m be Nat st n <= m holds dist(S1.m,g1) < p;
    thus for p be Real st 0 < p
    ex n be Nat st for m be Nat st n <= m holds |.seq.m - g.| < p
    proof
      let p be Real;
      assume 0 < p; then
      consider n be Nat such that
      A6: for m be Nat st n <= m holds dist(S1.m,g1) < p by A5;
      A7: for m be Nat st n <= m holds |.seq.m - g.| < p
      proof
        let m be Nat;
        assume
        A8: n <= m;
        set s = seq.m;
        set s1 = S1.m;
        dist(s1,g1) = |.s - g.| by A1,TOPMETR:11;
        hence thesis by A6,A8;
      end;
      take n;
      thus thesis by A7;
    end;
  end;
