reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of NAT;

theorem Th2:
  seq1 is convergent & (ex k st for n st k <= n holds seq2.n = seq1
  .n) implies seq2 is convergent
proof
  assume that
A1: seq1 is convergent and
A2: ex k st for n st k <= n holds seq2.n = seq1.n;
  consider k such that
A3: for n st n >= k holds seq2.n = seq1.n by A2;
  consider g such that
A4: for r st r > 0 ex m st for n st n >= m holds dist((seq1.n) , g) < r
  by A1;
  take h = g;
  let r;
  assume r > 0;
  then consider m1 be Nat such that
A5: for n st n >= m1 holds dist((seq1.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((seq1.n) , g) < r by A5;
    hence dist((seq2.n) , h) < r by A3,A8;
  end;
  now
    assume
A9: k <= m1;
    take m = m1;
    let n;
    assume
A10: n >= m;
    then seq2.n = seq1.n by A3,A9,XXREAL_0:2;
    hence dist((seq2.n) , h) < r by A5,A10;
  end;
  hence thesis by A6;
end;
