reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

theorem Th17:
  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,SEQ_2:def 7;
    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;
    m in NAT by ORDINAL1:def 12;
    then seq1.m=seq.(Nseq.m) by A3,FUNCT_2:15;
    hence |.(seq1.m)-(lim seq).|<p by A5,A7;
  end;
  seq1 is convergent by A1,A2,Th16;
  hence thesis by A4,SEQ_2:def 7;
end;
