Prove put-call parity for simple barrier options:

C_D/O + C_D/I − P_D/O − P_D/I = S − Ee^−r(T−t),
where C_D/O is a European down-and-out call, C_D/I is a European down and-in call, P_D/O is a European down-and-out put and PD/I is a European down-and-in put, all with expiry at time T and exercise price E.