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;
