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
  seq1 is convergent & (ex k st for n st n >= k holds seq2.n = seq1.n)
  implies lim seq1 = lim seq2
proof
  assume that
A1: seq1 is convergent and
A2: ex k st for n st n >= k holds seq2.n = seq1.n;
  consider k such that
A3: for n st n >= k holds seq2.n = seq1.n by A2;
A4: now
    let r;
    assume r > 0;
    then consider m1 be Nat such that
A5: for n st n >= m1 holds dist((seq1.n) , (lim seq1)) < r by A1,Def2;
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) , (lim seq1)) < r by A5;
      hence dist((seq2.n) , (lim seq1)) < 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) , (lim seq1)) < r by A5,A10;
    end;
    hence ex m st for n st n >= m holds dist((seq2.n) , (lim seq1)) < r by A6;
  end;
  seq2 is convergent by A1,A2,Th2;
  hence thesis by A4,Def2;
end;
