I read some facts about Lucas sequences, but they were not proven, can you help me?

Let P and Q be integers. Define $\displaystyle u_{n+1} = Pu_n - Qu_{n-1}$ with $\displaystyle u_0 = 0, u_1 = 1$.

Similar define $\displaystyle v_{n+1} = Pv_n - Qv_{n-1}$ with $\displaystyle v_0 = 2, v_1 = P$.

Show:

(1) $\displaystyle 2u_n = Pu_{n-1} + v_{n-1}$ (Managed to prove by induction myself)

(2) $\displaystyle 2v_n = (P^2-4Q)u_{n-1} + Pv_{n-1}$

(3) $\displaystyle u_{2n} = u_{n}v_{n}$

(4) $\displaystyle v_{2n} = v_{n}^2 - 2Q^{n}$

(5) $\displaystyle u_{n+m} = u_{n}u_{m+1} -Qu_{m}u_{n-1} = (u_{n}v_{m} + u_{m}v_{n})/2$

(6) $\displaystyle v_{n+m} = v_{n}v_{m} -Q^{m}v_{n-m}$

I would be thankful for any proven statement .