
theorem
  for n,k being Nat st (n >= k & k > 1) or (k = 1 & n > k) holds
  [\ tau to_power k * Fib n + 1/2 /] = Fib (n + k)
  proof
    let n,k be Nat;
    set tb = tau_bar;
    set tk = tau to_power k;
    set tbk = tau_bar to_power k;
    set ts = tau to_power (n+k);
    set tbs = tau_bar to_power (n+k);
    set tn = tau to_power n;
    set tbn = tau_bar to_power n;
    assume A1: (n >= k & k > 1) or (k = 1 & n > k);
A2: tk * Fib n = tk * ((tn - tbn) / sqrt 5) by FIB_NUM:7
    .= (tk * (tn - tbn)) / sqrt 5 by XCMPLX_1:74
    .= ((tk * tn - tbs) + (tbs - tk * tbn)) / sqrt 5
    .= ((ts - tbs) + (tbs - tk * tbn)) / sqrt 5 by Th2
    .= (ts - tbs) / sqrt 5 + (tbs - tk * tbn) / sqrt 5 by XCMPLX_1:62
    .= Fib (n+k) + (tbs - tk * tbn) / sqrt 5 by FIB_NUM:7
    .= Fib (n+k) + (tbn * tbk - tk * tbn) / sqrt 5 by Th2
    .= Fib (n+k) + (( - tbn) * (tk - tbk)) / sqrt 5
    .= Fib (n+k) + (- tbn) * ((tk - tbk) / sqrt 5) by XCMPLX_1:74
    .= Fib (n+k) + (- tbn) * Fib k by FIB_NUM:7
    .= Fib (n+k) - tbn * Fib k;
A3: tk * Fib n + 1/2 >= Fib (n+k)
    proof
      tbn * Fib k <= 1/2
      proof
        per cases;
        suppose A4: n is even;
          consider m being Nat such that A5: n = k + m by A1,NAT_1:10;
A6:       sqrt 5 > 0 by SQUARE_1:25;
          set tbm = tau_bar to_power m;
          tbm * ((-1) to_power k - tb to_power (2*k)) <= sqrt 5 / 2
          proof
            per cases;
            suppose A7: k is even;
              tb to_power (2*k) > 0 by Th6; then
A8:           - tb to_power (2*k) + 1 < 0 + 1 by XREAL_1:6;
              tb to_power (2*k) < 1 by Th8,A1,XXREAL_0:2; then
              - tb to_power (2*k) > - 1 by XREAL_1:24; then
A9:           - tb to_power (2*k) + 1 > - 1 + 1 by XREAL_1:6;
              m is even by A4,A5,A7; then
A10:           tbm > 0 by Th6;
              tbm <= sqrt 5 / 2
              proof
A11:             sqrt 5 > 2 by SQUARE_1:20,27;
                per cases;
                suppose A12: m = 0;
                  sqrt 5 / 2 >= 2 / 2 by A11,XREAL_1:72;
                  hence thesis by A12,POWER:24;
                end;
                suppose A13: m > 0;
                  sqrt 5 >= 1 by A11,XXREAL_0:2; then
                  sqrt 5 / 2 >= 1 / 2 by XREAL_1:72;
                  hence thesis by A13,Th8,XXREAL_0:2;
                end;
              end; then
              |.tbm.| <= sqrt 5/2 by A10,ABSVALUE:def 1; then
A14:           |.tbm.| * (1 - tb to_power (2*k)) <= (sqrt 5/2) * 1
                by A8,A9,XREAL_1:66;
              tbm * (1 - tb to_power (2*k)) <= |.tbm.| * (1 -tb to_power (2*k))
                by A9,ABSVALUE:4,XREAL_1:64; then
              tbm * (1 - tb to_power (2*k)) <= sqrt 5 / 2 by A14,XXREAL_0:2;
              hence thesis by A7,FIB_NUM2:3;
            end;
            suppose A15: k is odd; then
A16:           m is odd by A4,A5;
              m <> 0 by A15,A4,A5; then
A17:           m >= 1 by NAT_1:14;
              per cases by A17,XXREAL_0:1;
              suppose A18: m = 1;
                tb * (- 1 - tb to_power (2*k)) <= sqrt 5 / 2
                proof
A19:               tb to_power (2*k) > 0 by Th6;
                  tb to_power (2*k) < 1 / 2 by Th8,A1; then
                  tb to_power (2*k) + 1 < 1 / 2 + 1 by XREAL_1:6; then
                  ((sqrt 5 - 1) / sqrt 5) * (1 + tb to_power (2*k)) <
                  (6 / 10) * (3 / 2) by Lm23,Lm24,A19,XREAL_1:98; then
                  (((sqrt 5 - 1) / sqrt 5) * (1 + tb to_power (2*k))) < 1 &
                  sqrt 5 > 0 by SQUARE_1:25,XXREAL_0:2; then
                  (((sqrt 5 - 1) / sqrt 5) * (1 + tb to_power (2*k))) *sqrt 5 <
                  1 * sqrt 5 by XREAL_1:68; then
                  ((sqrt 5-1) * (1/sqrt 5)) * (1+tb to_power (2*k)) * sqrt 5 <
                  sqrt 5 by XCMPLX_1:99; then
                  (sqrt 5 - 1)* (1+tb to_power (2*k)) * (sqrt 5 *(1/sqrt 5)) <
                  sqrt 5; then
                  (sqrt 5-1)* (1+tb to_power (2*k))*(sqrt 5/sqrt 5) < 1*sqrt 5
                  & sqrt 5 > 0 by SQUARE_1:25,XCMPLX_1:99; then
                  (sqrt 5-1)*(1+tb to_power (2*k))*1<sqrt 5 by XCMPLX_1:60;then
                  ((1 - sqrt 5) * ( - 1 - tb to_power (2*k))) / 2 < sqrt 5 / 2
                    by XREAL_1:74;
                  hence thesis by FIB_NUM:def 2;
                end; then
                tbm * (- 1 - tb to_power (2*k)) <= sqrt 5 / 2 by A18;
                hence thesis by A15,FIB_NUM2:2;
              end;
              suppose A20: m > 1;
A21:             tbm < 0 by A16,Th7;
A22:             tbm * (-1-tb to_power (2*k)) = -tbm * (1+tb to_power (2*k));
A23:             tb to_power (2*k) > 0 by Th6;
                tb to_power (2*k) <= 1 / 2 by A1,Th8; then
A24:             tb to_power (2*k) + 1 <= 1/2 + 1 by XREAL_1:6;
                tbm > -1/2 by Th14,A20; then
                - tbm < - -1/2 by XREAL_1:24; then
                (-tbm) * (1 + tb to_power (2*k)) <= (1 / 2) * (3 / 2)
                  by A21,A23,A24,XREAL_1:66; then
                -tbm * (1 + tb to_power (2*k)) <= sqrt 5 / 2 by Lm25,XXREAL_0:2
;
                hence thesis by A15,A22,FIB_NUM2:2;
              end;
            end;
          end; then
          tbm*(tk*tbk-tb to_power (k+k)) <= sqrt 5/2 by Lm3,NEWTON:7; then
          tbm * (tk * tbk - tbk * tbk) <= sqrt 5 / 2 by Th2; then
          ((tbm * tbk) * (tk - tbk)) / sqrt 5 <= (sqrt 5 / 2) / sqrt 5
            by A6,XREAL_1:72; then
          (tbm*tbk) * ((tk - tbk) / sqrt 5) <= (sqrt 5 / 2)/sqrt 5
            by XCMPLX_1:74; then
          (tbm * tbk) * Fib k <= (sqrt 5 / 2) / sqrt 5 by FIB_NUM:7; then
          tbn * Fib k <= (sqrt 5 / 2) / sqrt 5 by A5,Th2; then
          tbn * Fib k <= (1 * sqrt 5) / (2 * sqrt 5) by XCMPLX_1:78;
          hence thesis by A6,XCMPLX_1:91;
        end;
        suppose n is odd; then
          tbn < 0 by Th7;
          hence thesis;
        end;
      end; then
      - tbn * Fib k >= - 1/2 by XREAL_1:24; then
      - tbn * Fib k + 1/2 >= - 1/2 + 1/2 by XREAL_1:6; then
      - tbn * Fib k + 1/2 + Fib (n+k) >= 0 + Fib (n+k) by XREAL_1:6;
      hence thesis by A2;
    end;
    tk * Fib n + 1/2 - 1 < Fib (n+k)
    proof
      tbn * Fib k > - 1/2
      proof
        per cases;
        suppose n is even; then
          tbn > 0 by Th6;
          hence thesis;
        end;
        suppose A25: n is odd;
          consider m being Nat such that A26: n = k + m by A1,NAT_1:10;
          set tbm = tau_bar to_power m;
          per cases;
          suppose A27: k is even; then
A28:         m is odd by A25,A26; then
A29:         tbm < 0 by Th7;
A30:         m >= 1 by A28,NAT_1:14;
            per cases by A30,XXREAL_0:1;
            suppose A31: m = 1;
              ((3-sqrt 5)/2) to_power k < 1 to_power k by Lm29,Lm28,A1,POWER:37
;
              then
              ((3 - sqrt 5) / 2) to_power k < 1; then
              - ((3-sqrt 5)/2) to_power k > - 1 by XREAL_1:24; then
A32:           - ((3-sqrt 5)/2) to_power k + 1 > - 1 + 1 by XREAL_1:6;
              ((3-sqrt 5)/2) to_power k > 0 by Lm28,POWER:34; then
A33:           - ((3-sqrt 5)/2) to_power k + 1 < 0 + 1 by XREAL_1:6;
A34:           1 - tb to_power (2*k) = (-1) to_power k - tb to_power (2*k)
                by A27,FIB_NUM2:3
              .= tk * tbk - tb to_power (k+k) by Lm3,NEWTON:7
              .= tk * tbk - tbk * tbk by Th2
              .= tbk * (tk - tbk);
              ((sqrt 5 - 1) / sqrt 5) * (1-((3-sqrt 5) / 2) to_power k) < 1 * 1
                by Lm26,Lm27,A32,A33,XREAL_1:98; then
              - ((sqrt 5-1)/sqrt 5) * (1-((3-sqrt 5)/2) to_power k) > -1
                by XREAL_1:24; then
              (- (sqrt 5-1) /sqrt 5) * (1-((3-sqrt 5)/2) to_power k) > -1; then
              ((- (sqrt 5-1)) / sqrt 5) * (1 - ((3-sqrt 5)/2) to_power k) > -1
                by XCMPLX_1:187; then
              ((1-sqrt 5)/sqrt 5) * (1 - tb to_power (2*k)) > - 1
                by Lm7,NEWTON:9; then
              (((1-sqrt 5) / sqrt 5) * (1-tb to_power (2*k))) / 2 > (-1) / 2
                by XREAL_1:74; then
              ((1-sqrt 5) /sqrt 5) * ((1-tb to_power (2*k)) / 2) > (-1)/2; then
              ((1-sqrt 5)*(1/sqrt 5)) * ((1-tb to_power (2*k))*(1/2)) > (-1)/2
                by XCMPLX_1:99; then
              tb*(1/sqrt 5)*(1-tb to_power (2*k)) > -1/2 by FIB_NUM:def 2; then
              tb*(1/sqrt 5) * (tbk * (tk-tbk)) > - 1/2 by A34; then
              tb * tbk * ((tk-tbk) * (1/sqrt 5)) > - 1/2; then
              tb * tbk * ((tk-tbk)/sqrt 5) > -1/2 by XCMPLX_1:99; then
              tb to_power 1 * tbk * ((tk-tbk)/sqrt 5) > - 1/2; then
              tb to_power (1+k) * ((tk-tbk) / sqrt 5) > - 1/2 by Th2;
              hence thesis by A31,A26,FIB_NUM:7;
            end;
            suppose m > 1; then
              tbm > - 1/2 by Th14; then
A35:           - tbm < - -1/2 by XREAL_1:24;
              sqrt 5 > 1 by SQUARE_1:18,27; then
              - sqrt 5 < - 1 by XREAL_1:24; then
A36:           (- sqrt 5) / 2 < (-1) / 2 by XREAL_1:74;
              tb to_power (2*k) > 0 & tb to_power (2*k) < 1/2
                by Th6,Th8,A1; then
              - tb to_power (2*k) < 0 & - tb to_power (2*k) > - 1/2
                by XREAL_1:24; then
              - tb to_power (2*k) + 1 < 0+1 & -tb to_power (2*k) + 1 > -1/2 + 1
                by XREAL_1:6; then
              (-tbm) * (1-tb to_power (2*k)) < (1/2) * 1
                by A35,A29,XREAL_1:98; then
              - -tbm * (1 - tb to_power (2*k)) > - 1/2 by XREAL_1:24; then
              tbm * ((-1) to_power k - tb to_power (2*k)) > - 1/2
                by A27,FIB_NUM2:3; then
              tbm * ((-1) to_power k - tb to_power (2*k)) > -sqrt 5/2
              by A36,XXREAL_0:2; then
              tbm * (tk * tbk - tb to_power (k+k)) > - sqrt 5 / 2
                by Lm3,NEWTON:7; then
              tbm * (tk * tbk - tbk * tbk) > - sqrt 5 / 2 by Th2; then
              (tbm * tbk) * (tk - tbk) > - sqrt 5 / 2; then
              tbn * (tk - tbk) > - sqrt 5 / 2 & sqrt 5 > 0
                by Th2,A26,SQUARE_1:25; then
              (tbn*(tk-tbk))/sqrt 5 > (-sqrt 5/2)/sqrt 5 by XREAL_1:74; then
              tbn * ((tk-tbk)/sqrt 5) > (-sqrt 5/2)/sqrt 5 by XCMPLX_1:74; then
              tbn * Fib k > ((-sqrt 5)/2)/sqrt 5 by FIB_NUM:7; then
              tbn * Fib k > ((-1) * sqrt 5) / (2 * sqrt 5) & sqrt 5 > 0
                by SQUARE_1:25,XCMPLX_1:78; then
              tbn * Fib k > (-1) / 2 by XCMPLX_1:91;
              hence thesis;
            end;
          end;
          suppose A37: k is odd; then
A38:         m is even by A25,A26;
            per cases;
            suppose A39: m = 0;
              per cases by A1;
              suppose k = 1;
                hence thesis by A1,A39,A26;
              end;
              suppose k > 1; then
                k >= 1+1 by NAT_1:13; then
                k = 2 or k > 2 by XXREAL_0:1; then
                k + 1 > 2 + 1 by A37,POLYFORM:5,XREAL_1:6; then
                k >= 3 by NAT_1:13; then
                k * 2 >= 3 * 2 by XREAL_1:64; then
                tb to_power (k*2) <= tb to_power (3*2) by Th11; then
A40:             tb to_power (2*k) + 1 <= 9-4*sqrt 5 + 1 by Lm9,XREAL_1:6;
                sqrt ((20/9) ^2) < sqrt 5 by SQUARE_1:27; then
                20 / 9 < sqrt 5 by SQUARE_1:def 2; then
                (20 / 9) * 9 < 9 * sqrt 5 by XREAL_1:68; then
                20-8*sqrt 5 < sqrt 5 + 8*sqrt 5 - 8*sqrt 5 by XREAL_1:9; then
                (20-8*sqrt 5) / 2 < sqrt 5 / 2 by XREAL_1:74; then
                tb to_power (2*k) + 1 < sqrt 5 / 2 by A40,XXREAL_0:2; then
                - (tb to_power (2*k) + 1) > - sqrt 5 / 2 by XREAL_1:24; then
                - tb to_power (2*k) + - 1 > - sqrt 5 / 2; then
                - tb to_power (2*k) + (-1) to_power k > - sqrt 5 / 2
                  by A37,FIB_NUM2:2; then
                - tb to_power (k+k) + tk * tbk >
                - sqrt 5 / 2 by Lm3,NEWTON:7; then
                - tbk * tbk + tk * tbk > - sqrt 5 / 2 by Th2; then
                tbk * (-tbk + tk) > -sqrt 5/2 & sqrt 5 > 0 by SQUARE_1:25; then
                (tbk * ( - tbk + tk)) / sqrt 5 >
                (- sqrt 5 / 2) / sqrt 5 by XREAL_1:74; then
                tbk*((tk-tbk)/sqrt 5) > (-sqrt 5/2)/sqrt 5 by XCMPLX_1:74; then
                tbk * Fib k > (- sqrt 5 / 2) / sqrt 5 by FIB_NUM:7; then
                tbk * Fib k > - (sqrt 5 / 2) / sqrt 5 by XCMPLX_1:187; then
                tbk * Fib k > - (sqrt 5 / sqrt 5) / 2 & sqrt 5 > 0
                by SQUARE_1:25,XCMPLX_1:48;
                hence thesis by A39,A26,XCMPLX_1:60;
              end;
            end;
            suppose m > 0; then
A41:           tbm > 0 & tbm < 1/2 by A38,Th6,Th8;
              sqrt ((3/2) ^2) < sqrt 5 by SQUARE_1:27; then
              3 / 2 < sqrt 5 by SQUARE_1:def 2; then
              (3 / 2) * 2 < sqrt 5 * 2 by XREAL_1:68; then
A42:           3 / (2 * 2) < (2 * sqrt 5) / (2 * 2) by XREAL_1:74;
A43:           tb to_power (2*k) < 1/2 & tb to_power (2*k) > 0
              by Th8,A1,Th6; then
              tb to_power (2*k) + 1 < 1/2 + 1 by XREAL_1:6; then
              tbm*(tb to_power (2*k)+1) < 1/2*(1/2+1) by A41,A43,XREAL_1:96;
 then
              tbm * (tb to_power (2*k) + 1) < sqrt 5/2 by A42,XXREAL_0:2; then
              - tbm * (tb to_power (2*k) + 1) > - sqrt 5/2 by XREAL_1:24; then
              tbm * (- tb to_power (2*k) +- 1) > - sqrt 5/2; then
              tbm * (- tb to_power (2*k) + (-1) to_power k) > - sqrt 5/2
              by A37,FIB_NUM2:2; then
              tbm * (- tb to_power (k+k) + tk * tbk) > - sqrt 5/2
              by Lm3,NEWTON:7; then
              tbm * (- tbk * tbk + tk * tbk) > - sqrt 5/2 by Th2; then
              (tbm * tbk) * (tk - tbk) > - sqrt 5/2; then
              tbn * (tk - tbk) > - sqrt 5 / 2 & sqrt 5 > 0
              by A26,Th2,SQUARE_1:25; then
              (tbn * (tk-tbk)) /sqrt 5 > (-sqrt 5/2)/sqrt 5 by XREAL_1:74; then
              tbn*((tk - tbk)/sqrt 5) > (-sqrt 5/2)/sqrt 5 by XCMPLX_1:74; then
              tbn * Fib k > ((-sqrt 5)/2)/sqrt 5 by FIB_NUM:7; then
              tbn * Fib k > ((-1) * sqrt 5) / (2 * sqrt 5) & sqrt 5 > 0
              by SQUARE_1:25,XCMPLX_1:78; then
              tbn * Fib k > (-1)/2 by XCMPLX_1:91;
              hence thesis;
            end;
          end;
        end;
      end; then
      - tbn * Fib k < - -1/2 by XREAL_1:24; then
      - tbn * Fib k + 1/2 < 1/2 + 1/2 by XREAL_1:6; then
      - tbn * Fib k + 1/2 - 1 < 1 - 1 by XREAL_1:9; then
      - tbn * Fib k + 1/2 - 1 + Fib (n+k) < 0 + Fib (n+k) by XREAL_1:6;
      hence thesis by A2;
    end;
    hence thesis by A3,INT_1:def 6;
  end;
