reserve k,m,n,p for Nat;
reserve x, a, b, c for Real;

theorem Th7:
  for n being Nat holds Fib(n) = ((tau to_power n) - (tau_bar
  to_power n))/(sqrt 5)
proof
  defpred P[Nat] means Fib($1) = ((tau to_power $1) - (tau_bar to_power $1))/(
  sqrt 5);
A1: tau_bar to_power 1 = tau_bar by POWER:25;
  tau_bar to_power 0 = 1 by POWER:24;
  then
  ((tau to_power 0) - (tau_bar to_power 0))/(sqrt 5) = (1 - 1)/(sqrt 5) by
POWER:24
    .= 0;
  then
A2: P[0] by PRE_FF:1;
A3: for k being Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be Nat;
    assume that
A4: P[k] and
A5: P[k+1];
    set a = tau to_power k, a1 = tau_bar to_power k, b = tau to_power (k+1),
b1 = tau_bar to_power (k+1), c = tau to_power (k+2), c1 = tau_bar to_power (k+2
    );
A6: c1 = tau_bar |^ (k + 2) by POWER:41
      .= (tau_bar |^ k) * (tau_bar |^ (1 + 1)) by NEWTON:8
      .= (tau_bar |^ k) * (tau_bar * (tau_bar |^ 1)) by NEWTON:6
      .= (tau_bar |^ k) * (tau_bar + 1) by Lm8
      .= (tau_bar |^ k * tau_bar) + (tau_bar |^ k) * 1
      .= (tau_bar |^ (k+1)) + (tau_bar |^ k) * 1 by NEWTON:6
      .= b1 + (tau_bar |^ k) by POWER:41
      .= a1 + b1 by POWER:41;
A7: c = (tau to_power 2) * (tau to_power k) by Lm12,POWER:27
      .= (tau to_power k) * (tau + 1) by Lm8,POWER:46
      .= (tau to_power k) * tau + (tau to_power k) * 1
      .= (tau to_power k) * (tau to_power 1) + a by POWER:25
      .= a + b by Lm12,POWER:27;
    Fib(k+2) = Fib((k + 1) + 1)
      .= (a - a1)/(sqrt 5) + (b - b1)/(sqrt 5) by A4,A5,PRE_FF:1
      .= (c - c1)/(sqrt 5) by A7,A6;
    hence thesis;
  end;
  tau - tau_bar = sqrt 5;
  then ((tau to_power 1) - (tau_bar to_power 1))/(sqrt 5) = (sqrt 5)/(sqrt 5)
  by A1,POWER:25
    .= Fib(1) by Lm10,PRE_FF:1,XCMPLX_1:60;
  then
A8: P[1];
  thus for n being Nat holds P[n] from FibInd(A2, A8, A3);
end;
