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

theorem
  for n,r being non zero Element of NAT st r <= n holds (Fib (n)) ^2 -
  Fib (n+r) * Fib (n-'r) = ((-1) |^(n-'r)) * (Fib (r)) ^2
proof
  set T = tau;
  set S = 1 / (sqrt 5);
  reconsider T as non zero Real by Th33;
  let n,r be non zero Element of NAT such that
A1: r <= n;
  set Y = n -' r;
  set X = n + r;
A2: X + Y = r + n + (n - r) by A1,XREAL_1:233
    .= (r + n + n) - r
    .= r + 2 * n -' r by NAT_1:12,XREAL_1:233
    .= r -' r + 2 * n by NAT_D:38
    .= 0 + 2 * n by XREAL_1:232
    .= 2 * n;
A3: X - Y = n + r - (n - r) by A1,XREAL_1:233
    .= 2 * r;
  set tyu = T to_power (-Y);
  set txu = T to_power (-X);
  set tnu = T to_power (-n);
  set ty = T to_power Y;
  set tx = T to_power X;
  set tn = T to_power n;
A4: -T <> 0 & - 1 = 2 * (-1) + 1;
  (Fib (n)) ^2 - Fib (X) * Fib (Y) = ((tn - (tau_bar to_power n)) / (sqrt
  5)) ^2 -Fib (X) * Fib (Y) by FIB_NUM:7
    .= ((tn - (tau_bar to_power n)) / (sqrt 5)) ^2 - ((tx - (tau_bar
  to_power X)) / (sqrt 5)) * (Fib (Y)) by FIB_NUM:7
    .= ((tn - (tau_bar to_power n)) / (sqrt 5)) ^2 - ((tx - (tau_bar
to_power X)) / (sqrt 5)) * ((ty - (tau_bar to_power Y)) / (sqrt 5)) by
FIB_NUM:7
    .= ((tn - (tau_bar to_power n)) * S) ^2 - ((tx - (tau_bar to_power X)) /
  (sqrt 5)) * ((ty - (tau_bar to_power Y)) / (sqrt 5)) by XCMPLX_1:99
    .= ((tn - (tau_bar to_power n)) * S) ^2 - ((tx - (tau_bar to_power X)) *
  S) * ((ty - (tau_bar to_power Y)) / (sqrt 5)) by XCMPLX_1:99
    .= ((tn - (tau_bar to_power n)) * S) ^2 - ((tx - (tau_bar to_power X)) *
  S) * ((ty - (tau_bar to_power Y)) * S) by XCMPLX_1:99
    .= (S ^2) * ((tn) ^2 - 2 * (tn) * ((-T) to_power (-1) to_power n) + ((-T
) to_power (-1) to_power n) ^2 - (tx - ((-T) to_power (-1) to_power X)) * (ty -
  ((-T) to_power (-1) to_power Y))) by Th34
    .= (S ^2) * ((tn) ^2 - 2 * (tn) * ((-T) to_power ((-1) * n)) + ((-T)
to_power (-1) to_power n) ^2 - (tx - ((-T) to_power (-1) to_power X)) * (ty - (
  (-T) to_power (-1) to_power Y))) by A4,Th6
    .= (S ^2) * ((tn) ^2 - 2 * tn * ((-T) to_power (- n)) + ((-T) to_power (
(-1) * n)) ^2 - (tx - ((-T) to_power (-1) to_power X)) * (ty - ((-T) to_power (
  -1) to_power Y))) by A4,Th6
    .= (S ^2) * (tn ^2 - 2 * tn * ((-T) to_power (- n)) + ((-T) to_power (-
n)) ^2 - (tx - ((-T) to_power ((-1) * X))) * (ty - ((-T) to_power (-1) to_power
  Y))) by Th35
    .= (S ^2) * (tn ^2 - 2 * tn * ((-T) to_power (- n)) + ((-T) to_power (-
  n)) ^2 - (tx - ((-T) to_power (- X))) * (ty - ((-T) to_power ((-1) * Y))))
by Th35
    .= (S ^2) * ((tn) ^2 - 2 * tn * (((-1) * T) to_power (-n)) + (((-1) * T)
  to_power (-n)) ^2 - tx * ty + tx * (((-1) * T) to_power (-Y)) + (((-1) * T)
to_power (-X)) * ty - (((-1) * T) to_power (-X)) * (((-1) * T) to_power (-Y)))
    .= (S ^2) * (tn ^2 - 2 * tn * (((-1) * T) to_power (-n)) + (((-1) * T)
  to_power (-n)) ^2 - tx * ty + tx * (((-1) * T) to_power (-Y)) + (((-1) * T)
to_power (-X)) * ty - (((-1) * T) to_power (-X)) * (((-1) to_power (-Y)) * tyu)
  ) by Th4
    .= (S ^2) * ((tn) ^2 - 2 * tn * (((-1) to_power (-n)) * tnu) + (((-1) *
T) to_power (-n)) ^2 - tx * ty + tx * (((-1) * T) to_power (-Y)) + (((-1) * T)
to_power (-X)) * ty - (((-1) * T) to_power (-X)) * (((-1) to_power (-Y)) * tyu)
  ) by Th4
    .= (S ^2) * (tn ^2 - 2 * tn * (((-1) to_power (-n)) * tnu) + (((-1)
to_power (-n)) * tnu) ^2 - tx * ty + tx * (((-1) * T) to_power (-Y)) + (((-1) *
  T) to_power (-X)) * ty - (((-1) * T) to_power (-X)) * (((-1) to_power (-Y)) *
  tyu)) by Th4
    .= (S ^2) * (tn ^2 - 2 * tn * (((-1) to_power (-n)) * tnu) + (((-1)
to_power (-n)) * tnu) ^2 - tx * ty + tx * (((-1) to_power (-Y)) * tyu) + (((-1)
* T) to_power (-X)) * ty - (((-1) * T) to_power (-X)) * (((-1) to_power (-Y)) *
  tyu)) by Th4
    .= (S ^2) * (tn ^2 - 2 * tn * (((-1) to_power (-n)) * tnu) + (((-1)
to_power (-n)) * tnu) ^2 - tx * ty + tx * (((-1) to_power (-Y)) * tyu) + (((-1)
to_power (-X)) * txu) * ty - (((-1) * T) to_power (-X)) * (((-1) to_power (-Y))
  * tyu)) by Th4
    .= (S ^2) * (tn ^2 - 2 * tn * (((-1) to_power (-n)) * tnu) + (((-1)
to_power (-n)) * tnu) ^2 - tx * ty + tx * (((-1) to_power (-Y)) * tyu) + (((-1)
to_power (-X)) * txu) * ty - (((-1) to_power (-X)) * txu) * (((-1) to_power (-Y
  )) * tyu)) by Th4
    .= (S ^2) * ((tn) ^2 - 2 * (tn * tnu) * ((-1) to_power (-n)) + ((-1)
to_power (-n)) ^2 * tnu ^2 - tx * ty + tx * ((-1) to_power (-Y)) * tyu + ((-1)
to_power (-X)) * txu * ty - (((-1) to_power (-X)) * txu * ((-1) to_power (-Y)))
  * tyu)
    .= (S ^2) * ((tn) ^2 - 2 * 1 * ((-1) to_power (-n)) + ((-1) to_power (-n
)) ^2 * tnu ^2 - tx * ty + tx * ((-1) to_power (-Y)) * tyu + ((-1) to_power (-X
  )) * txu * ty - (((-1) to_power (-X)) * txu * ((-1) to_power (-Y))) * tyu)
by Th10
    .= (S ^2) * (tn ^2 - 2 * ((-1) to_power (-n)) + 1 * tnu ^2 - tx * ty +
  tx * tyu * ((-1) to_power (-Y)) + ((-1) to_power (-X)) * txu * ty - (((-1)
  to_power (-X)) * ((-1) to_power (-Y)) * txu) * tyu) by Th7
    .= (S ^2) * ((tn) ^2 - 2 * ((-1) to_power (-n)) + tnu ^2 - tx * ty + tx
  * (1 / ty) * ((-1) to_power (-Y)) + ((-1) to_power (-X)) * txu * ty - (((-1)
  to_power (-X)) * ((-1) to_power (-Y)) * txu) * tyu) by Th33,POWER:28
    .= (S ^2) * ((tn) ^2 - 2 * ((-1) to_power (-n)) + tnu ^2 - tx * ty + (tx
  / ty) * ((-1) to_power (-Y)) + ((-1) to_power (-X)) * txu * ty - (((-1)
  to_power (-X)) * ((-1) to_power (-Y)) * txu) * tyu) by XCMPLX_1:99
    .= (S ^2) * (tn ^2 - 2 * ((-1) to_power (-n)) + tnu ^2 - tx * ty + (T
to_power (X-Y)) * ((-1) to_power (-Y)) + ((-1) to_power (-X)) * txu * ty - (((-
  1) to_power (-X)) * ((-1) to_power (-Y)) * txu) * tyu) by Th33,POWER:29
    .= (S ^2) * ((tn) ^2 - 2 * ((-1) to_power (-n)) + tnu ^2 - (T to_power (
X+Y)) + (T to_power (X-Y)) * ((-1) to_power (-Y)) + ((-1) to_power (-X)) * (ty
* txu) - (((-1) to_power (-X)) * ((-1) to_power (-Y))) * (txu * tyu)) by Th33,
POWER:27
    .= (S ^2) * (tn ^2 - 2 * ((-1) to_power (-n)) + tnu ^2 - (T to_power (X+
Y)) + (T to_power (X-Y)) * ((-1) to_power (-Y)) + ((-1) to_power (-X)) * (ty *
  (1 / tx)) - (((-1) to_power (-X)) * ((-1) to_power (-Y))) * (txu * tyu)) by
Th33,POWER:28
    .= (S ^2) * (tn ^2 - 2 * ((-1) to_power (-n)) + tnu ^2 - (T to_power (X+
Y)) + (T to_power (X-Y)) * ((-1) to_power (-Y)) + ((-1) to_power (-X)) * (ty /
  tx) - (((-1) to_power (-X)) * ((-1) to_power (-Y))) * ((txu) * (tyu))) by
XCMPLX_1:99
    .= (S ^2) * (tn ^2 - 2 * ((-1) to_power (-n)) + tnu ^2 - (T to_power (X+
  Y)) + (T to_power (X-Y)) * ((-1) to_power (-Y)) + ((-1) to_power (-X)) * (T
to_power (Y-X)) - (((-1) to_power (-X)) * ((-1) to_power (-Y))) * (txu * tyu))
  by Th33,POWER:29
    .= (S ^2) * (tn ^2 - 2 * ((-1) to_power (-n)) + tnu ^2 - (T to_power (X+
  Y)) + (T to_power (X-Y)) * ((-1) to_power (-Y)) + ((-1) to_power (-X)) * (T
  to_power (Y-X)) - ((-1) to_power (-X-Y)) * (txu * tyu)) by Th8
    .= (S ^2) * ((tn) ^2 - 2 * ((-1) to_power (-n)) + tnu ^2 - (T to_power (
2 * n)) + (T to_power (2 * r)) * ((-1) to_power (-Y)) + ((-1) to_power (-X)) *
(T to_power (-2 * r)) - ((-1) to_power (-2 * n)) * (T to_power (-2 * n))) by A2
,Th8
    .= (S ^2) * ((tn) ^2 - 2 * ((-1) to_power (-n)) + tnu ^2 - (T to_power (
2 * n)) + (T to_power (2 * r)) * ((-1) to_power (-Y)) + ((-1) to_power (-X)) *
  (T to_power (-2 * r)) - 1 * (T to_power (-2 * n))) by Th9
    .= (S ^2) * (tn to_power 2 - 2 * ((-1) to_power (-n)) + tnu ^2 - (T
  to_power (2 * n)) + (T to_power (2 * r)) * ((-1) to_power (-Y)) + ((-1)
to_power (-X)) * (T to_power (-2 * r)) - 1 * (T to_power (-2 * n))) by POWER:46
    .= (S ^2) * (T to_power (2 * n) - 2 * ((-1) to_power (-n)) + tnu ^2 - (T
  to_power (2 * n)) + (T to_power (2 * r)) * ((-1) to_power (-Y)) + ((-1)
to_power (-X)) * (T to_power (-2 * r)) - (T to_power (-2 * n))) by Th33,
POWER:33
    .= (S ^2) * (T to_power (2 * n) - 2 * ((-1) to_power n) + tnu ^2 - (T
  to_power (2 * n)) + (T to_power (2 * r)) * ((-1) to_power (-Y)) + ((-1)
  to_power (-X)) * (T to_power (-2 * r)) - (T to_power (-2 * n))) by Th11
    .= (S ^2) * (T to_power (2 * n) - (T to_power (2 * n)) - 2 * ((-1)
  to_power n) + (tnu) ^2 + (T to_power (2 * r)) * ((-1) to_power Y) + ((-1)
  to_power (-X)) * (T to_power (-2 * r)) - (T to_power (-2 * n))) by Th11
    .= (S ^2) * (- 2 * ((-1) to_power n) + tnu ^2 + (T to_power (2 * r)) * (
(-1) to_power Y) + ((-1) to_power X) * (T to_power (-2 * r)) - (T to_power (2 *
  (-n)))) by Th11
    .= (S ^2) * (- 2 * ((-1) to_power n) + tnu ^2 + (T to_power (2 * r)) * (
(-1) to_power Y) + ((-1) to_power X) * (T to_power (-2 * r)) - (T to_power (-n)
  to_power 2)) by Th33,POWER:33
    .= (S ^2) * (- 2 * ((-1) to_power n) + (T to_power (2 * r)) * ((-1)
to_power Y) + ((-1) to_power X) * (T to_power (-2 * r)) + (tnu) ^2 - tnu ^2)
by POWER:46
    .= (S ^2) * (-2 * ((-1) to_power (n-'r+r)) + ((-1) to_power Y) * (T
to_power (2 * r)) + ((-1) to_power (2 * r + Y)) * (T to_power (-2 * r))) by A3
    .= (S ^2) * (- 2 * (((-1) to_power r) * ((-1) to_power (n -' r))) + ((-1
  ) to_power Y) * (T to_power (2 * r)) + ((-1) to_power (2 * r + Y)) * (T
  to_power (-2 * r))) by Th5
    .= (S ^2) * (- 2 * (((-1) to_power r) * ((-1) to_power (n -' r))) + ((-1
  ) to_power (n-'r)) * (T to_power (2 * r)) + (((-1) to_power (2 * r)) * ((-1)
  to_power (n -'r))) * (T to_power (-2 * r))) by Th5
    .= (S ^2) * (-2 * (((-1) to_power r)) + T to_power (2 * r) + T to_power
  (-2 * r) * ((-1) to_power (2 * r))) * ((-1) to_power (n-'r))
    .= (S ^2) * (-2 * (((-1) to_power r)) + T to_power (2 * r) + T to_power
  (-2 * r) * 1) * ((-1) to_power (n-'r)) by Th3
    .= (S ^2) * (T to_power (2 * r) - 2 * ((-1) to_power r) + T to_power (2
  * (-r))) * ((-1) to_power (n-'r))
    .= (S ^2) * (T to_power r to_power 2 - 2 * ((-1) to_power r) + T
  to_power ((-r) * 2)) * ((-1) to_power (n-'r)) by Th33,POWER:33
    .= (S ^2) * ((T to_power r) ^2 - 2 * ((-1) to_power r) + T to_power ((-r
  ) * 2)) * ((-1) to_power (n-'r)) by POWER:46
    .= ((-1) to_power (n-'r)) * ((S^2) * ((T to_power r) ^2 - 2 * ((-1)
  to_power r) + T to_power (-r) to_power 2)) by Th33,POWER:33
    .= ((-1) to_power (n-'r)) * ((S^2) * ((T to_power r) ^2 - 2 * ((-1)
  to_power r) + (T to_power (-r)) ^2)) by POWER:46
    .= ((-1) to_power (n-'r)) * ((S^2) * (((tau to_power r) - (tau_bar
  to_power r)) ^2)) by Th37
    .= ((-1) to_power (n-'r)) * (((tau to_power r) - (tau_bar to_power r)) *
  S) ^2
    .= ((-1) to_power (n-'r)) * (((tau to_power r) - (tau_bar to_power r)) /
  (sqrt 5)) ^2 by XCMPLX_1:99
    .= ((-1) |^(n-'r)) * (((T to_power r) - (tau_bar to_power r)) / (sqrt 5)
  ) ^2 by POWER:41
    .= ((-1) |^(n-'r)) * (Fib (r)) ^2 by FIB_NUM:7;
  hence thesis;
end;
