theorem Th25:
  seq is non-increasing & seq1 is subsequence of seq implies seq1
  is non-increasing
proof
  assume that
A1: seq is non-increasing 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+1)))<=(seq.(Nseq.n)) by A1,Th8;
  then (seq*Nseq).(n+1)<=(seq.(Nseq.n)) by FUNCT_2:15;
  hence thesis by A3,FUNCT_2:15,A4;
end;
