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 Th10:
  seq1 is subsequence of seq & seq is convergent implies lim seq1= lim seq
proof
  assume that
A1: seq1 is subsequence of seq and
A2: seq is convergent;
  consider Nseq such that
A3: seq1 = seq * Nseq by A1,VALUED_0:def 17;
A4: now
    let r;
    assume r > 0;
    then consider m1 being Nat such that
A5: for n being Nat st n >= m1 holds dist((seq.n), (lim seq)) < r
        by A2,BHSP_2:def 2;
    take m = m1;
    let n be Nat such that
A6: n >= m;
A7: n in NAT by ORDINAL1:def 12;
    Nseq.n >= n by SEQM_3:14;
    then
A8: Nseq.n >= m by A6,XXREAL_0:2;
    seq1.n = seq.(Nseq.n) by A3,FUNCT_2:15,A7;
    hence dist((seq1.n), (lim seq)) < r by A5,A8;
  end;
  seq1 is convergent by A1,A2;
  hence thesis by A4,BHSP_2:def 2;
end;
