reserve a,b,x,y for Real,
  i,k,n,m for Nat;
reserve p,w for Real;
reserve seq for Real_Sequence;

theorem Th2:
  seq is increasing & rng seq c= NAT implies n <= seq.n
proof
  defpred P[Nat] means $1 <= seq.$1;
  assume that
A1: seq is increasing and
A2: rng seq c= NAT;
A3: for k st P[k] holds P[k+1]
  proof
    let k such that
A4: k <= seq.k;
    k+1 in NAT;
    then k+1 in dom seq by FUNCT_2:def 1;
    then seq.(k+1) in rng seq by FUNCT_1:def 3;
    then reconsider k9 = seq.(k+1) as Element of NAT by A2;
    seq.k < seq.(k+1) by A1,SEQM_3:def 6;
    then k < k9 by A4,XXREAL_0:2;
    hence thesis by NAT_1:13;
  end;
  0 in NAT;
  then 0 in dom seq by FUNCT_2:def 1;
  then seq.0 in rng seq by FUNCT_1:def 3;
  then
A5: P[0] by A2;
  for n holds P[n] from NAT_1:sch 2(A5,A3);
  hence thesis;
end;
