
theorem
  for n,k being Nat st n >= 4 & k >= 1 & n > k & n is odd holds
  Lucas (n+k) = [\ tau to_power k * Lucas n + 1 /]
  proof
    let n,k be Nat;
    assume A1: n >= 4 & k >= 1 & n > k & n is odd;
    set tb = tau_bar;
    set tk = tau to_power k;
    set tn = tau to_power n;
    set tbn = tau_bar to_power n;
A2:  sqrt 5 > 1 by SQUARE_1:18,27;
A3: tau to_power k * Lucas n + 1 >= Lucas (n+k)
    proof
      tk * tbn + 1 >= tb to_power (n+k)
      proof
        consider m being Nat such that A4: n = k + m by A1,NAT_1:10;
A5:     m is non zero Nat by A1,A4; then
A6:     m >= 1 by NAT_1:14;
A7:     ((1-sqrt 5) to_power m * (-1) to_power k) / 2 to_power m + 1 >=
        (1-sqrt 5) to_power (2*k+m) / 2 to_power (2*k+m)
        proof
          per cases;
          suppose A8: k is even; then
A9:         m is odd by A1,A4;
A10:         2 to_power m > 0 by POWER:34;
A11:         ((1-sqrt 5) to_power m * (-1) to_power k) / 2 to_power m + 1 =
            ((1-sqrt 5) to_power m *1) / 2 to_power m + 1 by A8,FIB_NUM2:3
            .= (( - ( - 1 + sqrt 5)) to_power m) / 2 to_power m +
            2 to_power m / 2 to_power m by A10,XCMPLX_1:60
            .= (((-1) * (sqrt 5 - 1)) to_power m + 2 to_power m) / 2 to_power m
              by XCMPLX_1:62
            .= ((-1) to_power m * (sqrt 5-1) to_power m +
            2 to_power m) / 2 to_power m by NEWTON:7
            .= (2 to_power m + (-1) * (sqrt 5-1) to_power m) / 2 to_power m
              by A9,FIB_NUM2:2
            .= (2 to_power m - (sqrt 5 - 1) to_power m) / 2 to_power m;
A12:          sqrt (3 ^2) > sqrt 5 & sqrt 5 > sqrt 1 by SQUARE_1:27; then
            3 > sqrt 5 & sqrt 5 > 1 by SQUARE_1:def 2; then
            3 - 1 > sqrt 5 - 1 & sqrt 5 - 1 > 1 - 1 by XREAL_1:9; then
            2 to_power m > (sqrt 5 - 1) to_power m by A5,POWER:37; then
A13:         2 to_power m - (sqrt 5-1) to_power m > (sqrt 5-1) to_power m -
            (sqrt 5-1) to_power m by XREAL_1:9;
A14:         (1-sqrt 5) to_power (2*k+m) / 2 to_power (2*k+m)
            = ((-1) * (-1+sqrt 5)) to_power (2*k+m) / 2 to_power (2*k+m)
            .= ((-1) to_power (2*k+m) * (-1+sqrt 5) to_power (2*k+m)) /
            2 to_power (2*k+m) by NEWTON:7
            .= ((-1) * (-1+sqrt 5) to_power (2*k+m)) / 2 to_power (2*k+m)
            by A9,FIB_NUM2:2
            .= (-1) * ((-1+sqrt 5) to_power (2*k+m) / 2 to_power (2*k+m))
            by XCMPLX_1:74;
            sqrt 5 - 1 > 1 - 1 by A12,XREAL_1:9; then
            (- 1 + sqrt 5) to_power (2*k+m) > 0 by POWER:34; then
            (1 - sqrt 5) to_power (2*k+m) / 2 to_power (2*k+m) <= 0 by A14;
            hence thesis by A13,A11;
          end;
          suppose A15: k is odd; then
A16:         m is even by A1,A4;
A17:         2 to_power m > 0 by POWER:34;
A18:         2 to_power (2*k) > 0 by POWER:34;
            m > 1 by A16,A6,POLYFORM:4,XXREAL_0:1; then
A19:         m - 1 is non zero Nat by NAT_1:20;
A20:         ((1-sqrt 5) to_power m * (-1) to_power k) / 2 to_power m + 1 =
            ((1-sqrt 5) to_power m * (-1)) / 2 to_power m + 1
              by A15,FIB_NUM2:2
            .= ((1-sqrt 5) to_power m *(-1)) / 2 to_power m +
            2 to_power m / 2 to_power m by A17,XCMPLX_1:60
            .= (((-1) * (-1+sqrt 5)) to_power m * (-1) +
            2 to_power m) / 2 to_power m by XCMPLX_1:62
            .= (((-1) to_power m) * ((-1+sqrt 5) to_power m) * (-1) +
            2 to_power m) / 2 to_power m by NEWTON:7
            .= (1 * ((-1 + sqrt 5) to_power m) * (-1) +
            2 to_power m) / 2 to_power m by A16,FIB_NUM2:3
            .= ((-((-1+sqrt 5) to_power m) + 2 to_power m) * 2 to_power (2*k))/
            (2 to_power m * 2 to_power (2*k)) by A18,XCMPLX_1:91
            .= (-((-1+sqrt 5) to_power m) * 2 to_power (2*k) +
            2 to_power m * 2 to_power (2*k)) / 2 to_power (m+2*k) by Th2
            .= (-((-1+sqrt 5) to_power m) * 2 to_power (2*k) +
            2 to_power (m+2*k)) / 2 to_power (m+2*k) by Th2
            .= (2 to_power (m+2*k) - (-1+sqrt 5) to_power m *
            2 to_power (2*k)) / 2 to_power (m+2*k);
A21:         (1-sqrt 5) to_power (2*k+m) / 2 to_power (2*k+m) =
             ((-1) * (-1+sqrt 5)) to_power (2*k+m) / 2 to_power (2*k+m)
            .= ((-1) to_power (2*k+m) * (-1+sqrt 5) to_power (2*k+m)) /
            2 to_power (2*k+m) by NEWTON:7
            .= (1 * (- 1 + sqrt 5) to_power (2*k+m)) / 2 to_power (2*k+m)
            by A16,FIB_NUM2:3
            .= ((sqrt 5 - 1) to_power (2*k+m)) / 2 to_power (2*k+m);
            2 to_power m - (sqrt 5 - 1) to_power m >= (sqrt 5 - 1) to_power m
            proof
              defpred P[Nat] means
              2 to_power ($1+1) - (sqrt 5 - 1) to_power ($1+1) >=
              (sqrt 5 - 1) to_power ($1+1);
A22:           2 to_power (1+1) - (sqrt 5 - 1) to_power (1+1) =
              2 ^2 - (sqrt 5-1) to_power 2 by POWER:46
              .= 4 - (sqrt 5-1) ^2 by POWER:46
              .= 4 - ((sqrt 5) ^2 - 2 * sqrt 5 * 1 + 1 ^2)
              .= 4 - (5 - 2 * sqrt 5 + 1) by SQUARE_1:def 2
              .= 2 * sqrt 5 - 2;
A23:           (sqrt 5 - 1) to_power (1+1) = (sqrt 5 - 1) ^2 by POWER:46
              .= (sqrt 5) ^2 - 2 * sqrt 5 * 1 + 1 ^2
              .= 5 - 2 * sqrt 5 + 1 by SQUARE_1:def 2
              .= 6 - 2 * sqrt 5;
              sqrt 5 >= sqrt (2 ^2) by SQUARE_1:27; then
              sqrt 5 >= 2 by SQUARE_1:def 2; then
              sqrt 5 * 4 >= 2 * 4 by XREAL_1:64; then
              sqrt 5 * 2 + sqrt 5 * 2 - 2 >= 6 + 2 - 2 by XREAL_1:9; then
              sqrt 5*2 + sqrt 5*2 -2-2*sqrt 5 >= 6 - 2*sqrt 5 by XREAL_1:9;
              then
A24:           P[1] by A22,A23;
A25:           for k being non zero Nat st P[k] holds P[k+1]
              proof
                let k be non zero Nat;
A26:             sqrt 5 - 1 > 1 - 1 by A2,XREAL_1:9;
                assume P[k]; then
                2 to_power (k+1) - (sqrt 5 - 1) to_power (k+1) + (sqrt 5 - 1)
                to_power (k+1) >= (sqrt 5 - 1) to_power (k+1) +
                (sqrt 5 - 1) to_power (k+1) by XREAL_1:6; then
                2 to_power (k+1) * (sqrt 5-1) >=
                2*(sqrt 5-1) to_power (k+1) * (sqrt 5-1) by A26,XREAL_1:64;
 then
                2 to_power (k+1) * (sqrt 5-1) >=
                2*((sqrt 5 - 1) to_power (k+1) * (sqrt 5-1)); then
                2 to_power (k+1) * (sqrt 5-1) >= 2*((sqrt 5 - 1) to_power (k+1)
                * (sqrt 5-1) to_power 1); then
A27:             2 to_power (k+1) * (sqrt 5-1) >=
                2 * (sqrt 5 - 1) to_power (k+1+1) by Th2,A26;
                sqrt (3 ^2) >= sqrt 5 by SQUARE_1:27; then
                3 >= sqrt 5 by SQUARE_1:def 2; then
                3 - 1 >= sqrt 5 - 1 by XREAL_1:9; then
                2 * 2 to_power (k+1) >= (sqrt 5-1) * 2 to_power (k+1)
                by XREAL_1:64; then
                2 to_power 1 * 2 to_power (k+1) >=
                (sqrt 5-1) * 2 to_power (k+1); then
                2 to_power (1+k+1) >= (sqrt 5-1) * 2 to_power (k+1)
                by Th2; then
                2 to_power (1+k+1) >= 2 * (sqrt 5 - 1) to_power (k+1+1)
                by A27,XXREAL_0:2; then
                2 to_power (1+k+1) - (sqrt 5 - 1) to_power (k+1+1)  >=
                2 * (sqrt 5 - 1) to_power (k+1+1)-(sqrt 5 - 1) to_power (k+1+1)
                by XREAL_1:9;
                hence thesis;
              end;
              for k being non zero Nat holds P[k] from NAT_1:sch 10(A24,A25);
              then 2 to_power (m-1+1) - (sqrt 5 - 1) to_power (m-1+1) >=
              (sqrt 5 - 1) to_power (m-1+1) by A19;
              hence thesis;
            end; then
            (2 to_power m - (sqrt 5 - 1) to_power m)* 2 to_power (2*k) >=
            (sqrt 5 - 1) to_power m * 2 to_power (2*k) by XREAL_1:64; then
            2 to_power m * 2 to_power (2*k)-
            (sqrt 5-1) to_power m * 2 to_power (2*k) >=
            (sqrt 5 - 1) to_power m * 2 to_power (2*k); then
            2 to_power (m+2*k) - (sqrt 5 - 1) to_power m * 2 to_power (2*k) >=
            (sqrt 5 - 1) to_power m * 2 to_power (2*k) by Th2; then
            (2 to_power (m + 2 * k) -
            (sqrt 5 - 1) to_power m * 2 to_power (2*k)) / 2 to_power (m+2*k) >=
            ((sqrt 5 - 1) to_power m * 2 to_power (2*k)) / 2 to_power (m+2*k)
              by XREAL_1:72; then
A28:         (2 to_power m * 2 to_power (2*k) -
            (sqrt 5 - 1) to_power m * 2 to_power (2*k)) / 2 to_power (m+2*k) >=
            ((sqrt 5 - 1) to_power m * 2 to_power (2*k)) / 2 to_power (m+2*k)
              by Th2;
A29:         sqrt 5 - 1 > 1 - 1 by A2,XREAL_1:9;
            sqrt (3 ^2) > sqrt 5 & sqrt 5 > sqrt 1 by SQUARE_1:27; then
            3 > sqrt 5 & sqrt 5 > 1 by SQUARE_1:def 2; then
A30:         3 - 1 > sqrt 5 - 1 & sqrt 5 - 1 > 1 - 1 by XREAL_1:9; then
A31:         (sqrt 5 - 1) to_power m > 0 by POWER:34;
            2 to_power (2*k) >= (sqrt 5-1) to_power (2*k)
            by A30,A1,POWER:37; then
            2 to_power (2*k) * (sqrt 5-1) to_power m >=
            (sqrt 5-1) to_power (2*k)*(sqrt 5-1) to_power m
            by A31,XREAL_1:64; then
            (2 to_power (2*k) * (sqrt 5-1) to_power m) / 2 to_power (2*k+m) >=
            ((sqrt 5-1) to_power (2*k) * (sqrt 5-1) to_power m) /
            2 to_power (2*k+m) by XREAL_1:72; then
            ((2 to_power m - (sqrt 5-1) to_power m) * 2 to_power (2*k)) /
            2 to_power (m+2*k) >=
            ((sqrt 5-1) to_power (2*k)*(sqrt 5-1) to_power m) /
            2 to_power (2*k+m) by A28,XXREAL_0:2; then
            (2 to_power m *2 to_power (2*k) -
            (sqrt 5-1) to_power m * 2 to_power (2*k)) / 2 to_power (m+2*k) >=
            ((sqrt 5-1) to_power (2*k+m)) /2 to_power (2*k+m) by Th2,A29;
            hence thesis by A20,A21,Th2;
          end;
        end;
        sqrt 5 - 1 > 1 - 1 by A2,XREAL_1:9; then
A32:     - (sqrt 5 - 1) < 0;
        ((1+sqrt 5)*(1-sqrt 5)) / 2 to_power 2 = ((1+sqrt 5)*(1-sqrt 5)) / 2^2
          by POWER:46
        .= (1 ^2 - (sqrt 5) ^2) / 4
        .= (1 - 5) / 4 by SQUARE_1:def 2
        .= - 1; then
        (-1) to_power k =
         ((1+sqrt 5) * (1-sqrt 5)) to_power k  / (2 to_power 2) to_power k
        by Th1
        .= ((1+sqrt 5)*(1-sqrt 5)) to_power k  / 2 to_power (2*k)
        by NEWTON:9; then
        ((1-sqrt 5) to_power m * (((1+sqrt 5) * (1-sqrt 5)) to_power k) /
        2 to_power (2*k)) / 2 to_power m + 1 >=
        (1-sqrt 5) to_power (2*k+m) /2 to_power (2*k+m) by A7,XCMPLX_1:74; then
        ((1-sqrt 5) to_power m * (((1+sqrt 5)*(1-sqrt 5)) to_power k)) /
        (2 to_power (2*k)*2 to_power m) + 1 >=
        (1-sqrt 5) to_power (2*k+m) / 2 to_power (2*k+m) by XCMPLX_1:78; then
        ((1-sqrt 5) to_power m * (((1+sqrt 5)*(1-sqrt 5)) to_power k)) /
        2 to_power (2*k+m) + 1 >=
        (1-sqrt 5) to_power (2*k+m) / 2 to_power (2*k+m) by Th2; then
        ((1-sqrt 5) to_power m * ((1+sqrt 5) to_power k *
        (1-sqrt 5) to_power k)) / 2 to_power (2*k+m) + 1 >=
        (1-sqrt 5) to_power (2*k+m) / 2 to_power (2*k+m) by NEWTON:7; then
        ((1-sqrt 5) to_power m *(1-sqrt 5) to_power k*(1+sqrt 5) to_power k) /
        2 to_power (2*k+m) + 1 >=
        (1-sqrt 5) to_power (2*k+m) / 2 to_power (2*k+m); then
        ((1-sqrt 5) to_power (m+k) *(1+sqrt 5) to_power k) /
        2 to_power (k+k+m) + 1 >=
        (1-sqrt 5) to_power (2*k+m) / 2 to_power (2*k+m) by Th2,A32; then
        ((1-sqrt 5) to_power n *(1+sqrt 5) to_power k) /2 to_power (k+n) + 1 >=
        ((1-sqrt 5)/2) to_power (k+k+m) by A4,Th1; then
        ((1-sqrt 5) to_power n *(1+sqrt 5) to_power k) /
        (2 to_power n * 2 to_power k) + 1 >=
        tb to_power (k+n) by Th2,A4,FIB_NUM:def 2; then
        ((1-sqrt 5) to_power n/2 to_power n) * ((1+sqrt 5) to_power k /
        (2 to_power k)) + 1 >= tb to_power (k+n) by XCMPLX_1:76; then
        (((1-sqrt 5)/2) to_power n) *((1+sqrt 5) to_power k / (2 to_power k))+1
        >= tb to_power (k+n) by Th1;
        hence thesis by Th1,FIB_NUM:def 1,def 2;
      end; then
      tk * tbn + 1 + tau to_power (n+k) >=
      tb to_power (n+k) + tau to_power (n+k) by XREAL_1:6; then
      tau to_power (n+k)+tk*tbn+1 >= tau to_power (n+k)+tb to_power (n+k); then
      tk*tn+tk*tbn+1 >= tau to_power (n+k)+tb to_power (n+k) by Th2;then
      tk * (tn + tbn) + 1 >= Lucas (n+k) by FIB_NUM3:21;
      hence thesis by FIB_NUM3:21;
    end;
    tau to_power k * Lucas n + 1 - 1 < Lucas (n+k)
    proof
      defpred P[Nat] means tau to_power $1 * Lucas n < Lucas (n+ $1);
      tbn < 0 by Th7,A1; then
      tbn * tau - tbn * tb < 0; then
      tbn * tau - tbn * tb to_power 1 < 0; then
      tbn * tau - tb to_power (n+1) < 0 by Th2; then
      tbn*tau - tb to_power (n+1) + (tau to_power (n+1) +tb to_power (n+1)) <
      0 + (tau to_power (n+1) + tb to_power (n+1)) by XREAL_1:6; then
      tbn * tau + tau to_power (n+1) < Lucas (n+1) by FIB_NUM3:21; then
      tbn * tau + tn * tau to_power 1 < Lucas (n+1) by Th2; then
      tbn * tau + tn * tau < Lucas (n+1); then
      (tbn + tn) * tau < Lucas (n+1); then
      Lucas n * tau < Lucas (n+1) by FIB_NUM3:21; then
A33:   P[1];
A34:   for m being non zero Nat st P[m] holds P[m + 1]
      proof
        let m be non zero Nat;
        assume P[m];
A35:     (1-sqrt 5) to_power (m+1) < (1+sqrt 5) to_power (m+1)
        proof
          reconsider s = m + 1 as Element of NAT by ORDINAL1:def 12;
          (1 - sqrt 5) to_power s <=
          |.(1 - sqrt 5) to_power s.| by ABSVALUE:4; then
A36:       (1-sqrt 5) to_power s <= (|.1-sqrt 5.|) to_power s by SERIES_1:2;
          sqrt 5 > sqrt 1 by SQUARE_1:27; then
          - sqrt 5 < - 1 by XREAL_1:24; then
A37:       - sqrt 5 + 1 < - 1 + 1 by XREAL_1:6; then
A38:       |.1 - sqrt 5.| = - (1 - sqrt 5) by ABSVALUE:def 1;
          - (1 - sqrt 5) = -1 + sqrt 5; then
          - (1 - sqrt 5) < 1 + sqrt 5 by XREAL_1:6; then
          (|.1 - sqrt 5.|) to_power s < (1 + sqrt 5) to_power s
            by A38,A37,POWER:37;
          hence thesis by A36,XXREAL_0:2;
        end;
        2 to_power (m + 1) > 0 by POWER:34; then
        ((1 - sqrt 5) to_power (m+1)) / (2 to_power (m+1)) <
        ((1 + sqrt 5) to_power (m+1))/(2 to_power (m+1)) by A35,XREAL_1:74;
 then
        ((1 - sqrt 5)/2) to_power (m+1) <
        ((1 + sqrt 5) to_power (m+1)) / (2 to_power (m+1)) by Th1; then
        tb to_power (m+1) < tau to_power (m+1) & tbn < 0
        by A1,Th7,Th1,FIB_NUM:def 1,def 2; then
        tb to_power (m+1) * tbn > tau to_power (m+1) * tbn by XREAL_1:69; then
        tb to_power (m+1) * tbn + tau to_power (n+m+1) >
        tau to_power (m+1) * tbn + tau to_power (n+m+1) by XREAL_1:6; then
        tb to_power (m+1+n) + tau to_power (n+m+1) >
        tau to_power (m+1) * tbn + tau to_power (n+m+1) by Th2; then
        tb to_power (m+1+n) + tau to_power (n+m+1) >
        tau to_power (m+1) * tbn + tn * tau to_power (m+1) by Th2; then
        Lucas (n+m+1) > tau to_power (m+1) * (tbn + tn) by FIB_NUM3:21;
        hence thesis by FIB_NUM3:21;
      end;
      for m being non zero Nat holds P[m] from NAT_1:sch 10(A33,A34);
      hence thesis by A1;
    end;
    hence thesis by A3,INT_1:def 6;
  end;
