
theorem FiboLeq:
  for k,m being Nat st
    Fib k < Fib (m + 1) & 1 < k holds
      Fib k <= Fib m
  proof
    let k,m be Nat;
    assume that
A1: Fib k < Fib (m + 1) and
A0: 1 < k;
    k < m+1 by A1,FIB_NUM2:45; then
    k <= m by NAT_1:13; then
    per cases by XXREAL_0:1;
    suppose k = m;
      hence thesis;
    end;
    suppose k < m;
      hence thesis by A0,FIB_NUM2:46;
    end;
  end;
