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 Th24:
  seq is non-decreasing & seq1 is subsequence of seq implies seq1
  is non-decreasing
proof
  assume that
A1: seq is non-decreasing and
A2: seq1 is subsequence of seq;
  let n;
  consider Nseq such that
A3: seq1=seq*Nseq by A2,VALUED_0:def 17;
A4: n in NAT by ORDINAL1:def 12;
  Nseq.n<=Nseq.(n+1) by Lm7;
  then (seq.(Nseq.n))<=(seq.(Nseq.(n+1))) by A1,Th6;
  then (seq*Nseq).n<=(seq.(Nseq.(n+1))) by FUNCT_2:15,A4;
  hence thesis by A3,FUNCT_2:15;
end;
