
theorem Th8:
  for X be ComplexNormSpace, seq,seq1 be sequence of X holds seq1
  is subsequence of seq & seq is convergent implies lim seq1=lim seq
proof
  let X be ComplexNormSpace;
  let seq,seq1 be sequence of X;
  assume that
A1: seq1 is subsequence of seq and
A2: seq is convergent;
  consider Nseq be increasing sequence of NAT such that
A3: seq1=seq*Nseq by A1,VALUED_0:def 17;
A4: now
    let p be Real;
    assume 0<p;
    then consider n1 be Nat such that
A5: for m be Nat st n1<=m holds ||.(seq.m)-(lim seq).||<p
    by A2,CLVECT_1:def 16;
    take n=n1;
    let m be Nat such that
A6: n<=m;
A7: m in NAT by ORDINAL1:def 12;
    m<=Nseq.m by SEQM_3:14;
    then
A8: n<=Nseq.m by A6,XXREAL_0:2;
    seq1.m=seq.(Nseq.m) by A3,FUNCT_2:15,A7;
    hence ||.(seq1.m)-(lim seq).||<p by A5,A8;
  end;
  seq1 is convergent by A1,A2,Th7;
  hence thesis by A4,CLVECT_1:def 16;
end;
