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

theorem Th44:
  for n being Nat st n > 1 holds Fib (n) < Fib (n+1)
proof
  defpred P[Nat] means Fib ($1) < Fib ($1 + 1);
  let n be Nat;
  assume n > 1;
  then
A1: n is non trivial by NAT_2:def 1;
A2: P[3] by Th22,Th23;
A3: for k being non trivial Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be non trivial Nat;
    assume
A4: P[k];
    assume P[k+1];
    then Fib (k) + Fib (k+1) < Fib (k+1) + Fib (k+2) by A4,XREAL_1:8;
    then Fib (k+2) < Fib (k+1) + Fib (k+2) by Th24;
    then Fib (k+2) < Fib (k+3) by Th25;
    hence thesis;
  end;
A5: P[2] by Th21,Th22;
  for n being non trivial Nat holds P[n] from FibInd2(A5,A2,A3);
  hence thesis by A1;
end;
