
theorem LTAUPOW:
  for n be Nat st 3 <= n holds tau to_power (n-2) < Fib(n)
  proof
    let n be Nat;
    defpred P[Nat] means tau to_power ($1 -2) < Fib($1);
    A1: for k being Nat st k>=3 holds
    (for i being Nat st i>=3 holds i<k implies P[i]) implies P[k]
    proof
      let k be Nat;
      assume
      A2: k>=3;
      assume
      A3: for i being Nat st i>=3 holds i<k implies P[i];
      LL:(3<= k & k <=3+1) or 4 < k by A2;
      now per cases by LL, NAT_1:9;
        case k=3;
          tau to_power 1 = tau;
          hence P[3] by FIB_NUM2:22,THTU2;
        end;
        case k=4;
          tau to_power (4-2)=tau to_power (0+2)
          .=tau to_power 0 + tau to_power (0+1) by FIB_NUM3:9
          .= 1 + tau by POWER:24; then
          tau to_power (4-2) < 1 +2 by THTU2,XREAL_1:6;
          hence P[4] by FIB_NUM2:23;
        end;
        case LC4A: 4<k; then
          k -4 in NAT by INT_1:5; then
          reconsider z = k -4 as Nat;
          4-3 < k -0 by XREAL_1:14,LC4A; then
          k -1 in NAT by INT_1:5; then
          reconsider x = k -1 as Nat;
          4-2 < k -0 by XREAL_1:14,LC4A; then
          k -2 in NAT by INT_1:5; then
          reconsider y = k -2 as Nat;
          4+1 <= k by INT_1:7,LC4A; then
          5-2 <= k-2 & 5-2 <= k - 1 by XREAL_1:13; then
          3 <= x & 3 <= y & x < k & y < k by XREAL_1:44; then
          tau to_power (x -2) < Fib(x) &
          tau to_power (y -2) < Fib(y) by A3; then
          tau to_power (z) +tau to_power (z+1)
            < Fib(x) + Fib(y) by XREAL_1:8; then
          tau to_power (z+2) < Fib(x) + Fib(y) by FIB_NUM3:9; then
          tau to_power (k-2) < Fib((y+1)+1) by PRE_FF:1;
          hence P[k];
        end;
      end;
      hence thesis;
    end;
    for k being Nat st k>=3 holds P[k] from NAT_1:sch 9(A1);
    hence thesis;
  end;
