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

theorem Th43:
  Fib (n) <= Fib (n + 1)
proof
  defpred P[Nat] means Fib ($1) <= Fib ($1 + 1);
A1: P[0] by PRE_FF:1;
A2: for k being Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be Nat;
    assume
A3: P[k];
    assume P[k+1];
    then Fib (k) + Fib (k+1) <= Fib (k+1) + Fib (k+2) by A3,XREAL_1:7;
    then Fib (k+2) <= Fib (k+1) + Fib (k+2) by Th24;
    then Fib (k+2) <= Fib (k+3) by Th25;
    hence thesis;
  end;
A4: P[1] by Th21,PRE_FF:1;
  for n being Nat holds P[n] from FIB_NUM:sch 1(A1,A4,A2);
  hence thesis;
end;
