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
  seq1 is convergent & lim seq1 = g & seq1 is_compared_to seq2 implies
  seq2 is convergent & lim seq2 = g
proof
  assume that
A1: seq1 is convergent & lim seq1 = g and
A2: seq1 is_compared_to seq2;
A3: now
    let r;
    assume r > 0;
    then
A4: r/2 > 0 by XREAL_1:215;
    then consider m1 being Nat such that
A5: for n being Nat st n >= m1 holds dist((seq1.n), g) < r/2
           by A1,BHSP_2:def 2;
    consider m2 such that
A6: for n st n >= m2 holds dist((seq1.n), (seq2.n)) < r/2 by A2,A4;
    reconsider m = m1 + m2 as Nat;
    take m;
    let n being Nat such that
A7: n >= m;
    m >= m2 by NAT_1:12;
    then n >= m2 by A7,XXREAL_0:2;
    then
A8: dist((seq1.n), (seq2.n)) < r/2 by A6;
A9: dist((seq2.n), g) <= dist((seq2.n), (seq1.n)) + dist((seq1.n), g) by
BHSP_1:35;
    m1 + m2 >= m1 by NAT_1:12;
    then n >= m1 by A7,XXREAL_0:2;
    then dist((seq1.n), g) < r/2 by A5;
    then dist((seq2.n), (seq1.n)) + dist((seq1.n), g) < r/2 + r/2 by A8,
XREAL_1:8;
    hence dist((seq2.n), g) < r by A9,XXREAL_0:2;
  end;
  then seq2 is convergent by BHSP_2:def 1;
  hence thesis by A3,BHSP_2:def 2;
end;
