Prove put-call parity for European compound options:

C_C + P_P − C_P − P_C = S − E₂e^−r(T2−t),

where C_C is a call on a call, C_P is a call on a put, P_C is a put on a call and P_P is a put on a put. The compound options have exercise price E₁ and expiry at time T₁ and the underlying calls and puts have exercise price E₂ and expiry at time T₂.