reserve c, c1, d for Real,
  k for Nat,
  n, m, N, n1, N1, N2, N3, N4, N5, M for Element of NAT,
  x for set;

theorem Th1: :: Problem 3.25
  for f being Real_Sequence, N being Nat
    st for n being Nat st n>= N holds f.n <= f.(n+1)
  for n,m being Nat st N <= n & n <= m
  holds f.n <= f.m
proof
  let f be Real_Sequence, N be Nat;
  assume
A1: for n being Nat st n >= N holds f.n <= f.(n+1);
  let n,m be Nat;
  defpred P[Nat] means f.n <= f.$1;
  assume
A2: n >= N;
A3: for m be Nat st m >= n & P[m] holds P[m+1]
  proof
    let m be Nat;
    assume that
A4: m >= n and
A5: f.n <= f.m;
    m in NAT & m >= N by A2,A4,ORDINAL1:def 12,XXREAL_0:2;
    then f.m <= f.(m+1) by A1;
    hence thesis by A5,XXREAL_0:2;
  end;
A6: P[n];
  for m be Nat st m >= n holds P[m] from NAT_1:sch 8(A6,A3);
  hence thesis;
end;
