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
  seq is bounded_below & seq is non-increasing implies for n holds lim
  seq <= seq.n
proof
  assume that
A1: seq is bounded_below and
A2: seq is non-increasing;
  let m;
  set seq1 = NAT --> seq.m;
  deffunc U(Nat) = seq.($1+m);
  consider seq2 such that
A3: for n holds seq2.n=U(n) from SEQ_1:sch 1;
A4: now
    let n;
    n in NAT by ORDINAL1:def 12;
    then seq1.n=seq.m & seq2.n=seq.(m+n) by A3,FUNCOP_1:7;
    hence seq2.n<=seq1.n by A2,SEQM_3:7;
  end;
  seq1.0=seq.m;
  then
A5: lim seq1=seq.m by Th25;
  for n being Nat holds seq2.n=U(n) by A3;
  then
A6: seq2=seq^\m by NAT_1:def 3;
  then lim seq2=lim seq by A1,A2,Th17;
  hence thesis by A1,A2,A5,A6,A4,SEQ_2:18;
end;
