reserve X for RealUnitarySpace,
  x, g, g1, h for Point of X,
  a, p, r, M, M1, M2 for Real,
  seq, seq1, seq2, seq3 for sequence of X,
  Nseq for increasing sequence of NAT,

  k, l, l1, l2, l3, n, m, m1, m2 for 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 being Nat st for n being Nat st n >= m holds ||.seq.n - g1.|| < r
        by A1,BHSP_2:9;
  now
    take g = g1;
    let r;
    assume r > 0;
    then consider m1 being Nat such that
A5: for n being Nat st n >= m1 holds ||.seq.n - g.|| < r by A4;
     reconsider m = m1 + k as Nat;
    take m;
    let n be Nat;
    assume
A6: n >= m;
    then consider m2 being 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 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 BHSP_2:9;
end;
