reserve a,b,n for Element of NAT;

theorem Th28:
  for n being Nat holds Fib(2*n) = Fib(n) * Lucas(n)
proof
  let n be Nat;
A1: n in NAT by ORDINAL1:def 12;
  Fib(n)*Lucas(n)= (((tau to_power n) - (tau_bar to_power n))/(sqrt 5))*
  Lucas(n) by FIB_NUM:7
    .=(((tau to_power n) - (tau_bar to_power n))/(sqrt 5)) * (tau to_power n
  + tau_bar to_power n) by Th21
    .=( tau to_power n + tau_bar to_power n)* ( tau to_power n - tau_bar
  to_power n)/(sqrt 5) by XCMPLX_1:74
    .=((tau to_power n)to_power 2 - (tau_bar to_power n)to_power 2)/(sqrt 5)
  by Th7
    .=((tau to_power n)to_power 2 - tau_bar to_power (2*n))/(sqrt 5) by A1,Th6
    .= (tau to_power (2*n) - tau_bar to_power (2*n))/(sqrt 5) by POWER:33
    .=Fib(2*n) by FIB_NUM:7;
  hence thesis;
end;
