reserve n,n1,m,m1,k for Nat;
reserve x,X,X1 for set;
reserve g,g1,g2,t,x0,x1,x2 for Complex;
reserve s1,s2,q1,seq,seq1,seq2,seq3 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f,f1,f2,h,h1,h2 for PartFunc of COMPLEX,COMPLEX;
reserve p,r,s for Real;
reserve Ns,Nseq for increasing sequence of NAT;

theorem Th18:
  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 p;
    assume 0<p;
    then consider n1 such that
A5: for m st n1<=m holds |.(seq.m)-(lim seq).|<p by A2,COMSEQ_2:def 6;
    take n=n1;
    let m such that
A6: n<=m;
    m<=Nseq.m by SEQM_3:14;
    then
A7: n<=Nseq.m by A6,XXREAL_0:2;
A8: m in NAT by ORDINAL1:def 12;
    seq1.m=seq.(Nseq.m) by A3,FUNCT_2:15,A8;
    hence |.(seq1.m)-(lim seq).|<p by A5,A7;
  end;
  seq1 is convergent by A1,A2,Th17;
  hence thesis by A4,COMSEQ_2:def 6;
end;
