Let $N_t$ be a Poisson process and $T_{N_t}=X_1+\ldots+X_{N_t}$ where $X_i$ has an exponential law ($E(\lambda)$). Let $A_t=t-T_{N_t-1}$ and $B_t=T_{N_t}-t$.
Show that for $x,y,t \geq 0$, $P(B_t \geq x , A_t \geq y)=\lambda \int_{x+y}^{\infty} P(X_1 \geq u)du$
My idea: $P(B_t \geq x , A_t \geq y)=P(B_t \geq x |A_t \geq y)P(A_t \geq y)$ but I don't know to calculate these probabilities.
Thank you