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
  seq is convergent & (ex k st seq = seq1 ^\k) implies seq1 is convergent
proof
  assume that
A1: seq is convergent and
A2: ex k st seq = seq1 ^\k;
  consider k such that
A3: seq = seq1 ^\k by A2;
  consider g1 such that
A4: for r st r > 0 ex m st for n st n >= m holds ||.seq.n - g1.|| < r by A1,Th9
;
  now
    take g = g1;
    let r;
    assume r > 0;
    then consider m1 be Nat such that
A5: for n st n >= m1 holds ||.seq.n - g.|| < r by A4;
    take m = m1 + k;
    let n;
    assume
A6: n >= m;
    then consider m2 be Nat such that
A7: n = m1 + k + m2 by NAT_1:10;
    reconsider m2 as Nat;
    n - k = m1 + m2 by A7;
    then consider l being Nat such that
A8: l = n - k;
    now
      assume l < m1;
      then l + k < m1 + k by XREAL_1:6;
      hence contradiction by A6,A8;
    end;
    then
A9: ||.seq.l - g.|| < r by A5;
    l + k = n by A8;
    hence ||.seq1.n - g.|| < r by A3,A9,NAT_1:def 3;
  end;
  hence thesis by Th9;
end;
