reserve n, k, r, m, i, j for Nat;

theorem
  for m, n st m >= n holds Fib(m) >= Fib(n)
proof
  let m, n;
  assume m >= n;
  then consider k be Nat such that
A1: m = n + k by NAT_1:10;
  for k, n being Nat holds Fib(n+k) >= Fib(n)
  proof
    defpred P[Nat] means for n being Nat holds Fib(n+$1) >= Fib(n);
A2: for k being Nat st P[k] holds P[k+1]
    proof
      let k;
      assume
A3:   P[k];
      let n;
      n + (k+1) = (n+k) + 1;
      then
A4:   Fib(n + (k+1)) >= Fib(n+k) by Th43;
      Fib(n+k) >= Fib(n) by A3;
      hence thesis by A4,XXREAL_0:2;
    end;
    let k;
    let n;
A5: P[0];
    for k holds P[k] from NAT_1:sch 2(A5, A2);
    hence thesis;
  end;
  hence thesis by A1;
end;
