reserve n,m,k for Nat;
reserve r,r1 for Real;
reserve f,seq,seq1 for Real_Sequence;
reserve x,y for set;
reserve e1,e2 for ExtReal;
reserve Nseq for increasing sequence of NAT;

theorem Th28:
  seq is bounded_below & seq1 is subsequence of seq implies seq1
  is bounded_below
proof
  assume that
A1: seq is bounded_below and
A2: seq1 is subsequence of seq;
  consider r1 such that
A3: for n holds r1<seq.n by A1;
  consider Nseq such that
A4: seq1=seq*Nseq by A2,VALUED_0:def 17;
  take r=r1;
  let n be Nat;
  n in NAT by ORDINAL1:def 12;
  then seq1.n=seq.(Nseq.n) by A4,FUNCT_2:15;
  hence r<seq1.n by A3;
end;
