reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem Th27:
  seq is divergent_to-infty & seq1 is subsequence of seq implies
  seq1 is divergent_to-infty
proof
  assume that
A1: seq is divergent_to-infty and
A2: seq1 is subsequence of seq;
  consider Ns be increasing sequence of NAT such that
A3: seq1=seq*Ns by A2,VALUED_0:def 17;
  let r;
  consider n such that
A4: for m st n<=m holds seq.m<r by A1;
  take n;
  let m;
  assume
A5: n<=m;
A6: m in NAT by ORDINAL1:def 12;
  m<=Ns.m by SEQM_3:14;
  then n<=Ns.m by A5,XXREAL_0:2;
  then seq.(Ns.m)<r by A4;
  hence thesis by A3,FUNCT_2:15,A6;
end;
