Is this random variable exponential?

Question

My problem:

Suppose $Y$ is a real random variable with $Y \sim Exp(\lambda)$ and $Z$ is indipendent from $Y$, with density and $Z \geq 0$.

Why can I say $\mathbb{P}(Y-Z \geq t|Y>Z)=e^{-\lambda t}$?

My attempt:

I actually do not know how to proceed.

Answer 1

$P(Y >t+s)=e^{-\lambda(t+s)}=e^{-\lambda t} P(Y>s)$ for each $s$. Integrating w.r.t. the distribution $F_Z$ of $Z$ and using indepedence this becomes $P(Y>t+Z)=e^{-\lambda t} P(Y>Z)$. Divide by $P(Y>Z)$ to finish the proof.