reserve X for RealUnitarySpace;
reserve x, y, z, g, g1, g2 for Point of X;
reserve a, q, r for Real;
reserve seq, seq1, seq2, seq9 for sequence of X;
reserve k, n, m, m1, m2 for Nat;

theorem Th2:
  seq is convergent & (ex k st for n st k <= n holds seq9.n = seq.n
  ) implies seq9 is convergent
proof
  assume that
A1: seq is convergent and
A2: ex k st for n st k <= n holds seq9.n = seq.n;
  consider k such that
A3: for n st n >= k holds seq9.n = seq.n by A2;
  consider g such that
A4: for r st r > 0 ex m st for n st n >= m holds dist((seq.n) , g) < r
  by A1;
  take h = g;
  let r;
  assume r > 0;
  then consider m1 such that
A5: for n st n >= m1 holds dist((seq.n) , h) < r by A4;
A6: now
    assume
A7: m1 <= k;
    take m = k;
    let n;
    assume
A8: n >= m;
    then n >= m1 by A7,XXREAL_0:2;
    then dist((seq.n) , g) < r by A5;
    hence dist((seq9.n) , h) < r by A3,A8;
  end;
  now
    assume
A9: k <= m1;
    take m = m1;
    let n;
    assume
A10: n >= m;
    then seq9.n = seq.n by A3,A9,XXREAL_0:2;
    hence dist((seq9.n) , h) < r by A5,A10;
  end;
  hence thesis by A6;
end;
