Ito isometry for stopping time

Question

I am trying to prove the following: for the progressively measurable stochastic process $H(t,\omega)$ and some stopping time $\tau$ for which $\mathbb{E} \int_0^\tau H(s)^2ds < \infty$, the following are true: $$ 1) \mathbb{E} \int_0^\tau H(s)dB(s)=0$$ $$2) \mathbb{E} {\left[\int_0^\tau H(s)dB(s)\right]}^2=\mathbb{E} \int_0^\tau H(s)^2ds$$ For the first one, I want to use the optional stopping theorem, but why is it justified? For the second, this looks like the Ito isometry, but I am unsure how to prove it.