Part 1 should start as: let $\displaystyle y\in f(f^{-1}(H))$. Then $\displaystyle y=f(x)$ for some $\displaystyle x\in f^{-1}(H)$ ...

Part 2 should start as: let $\displaystyle y\in H$. Now, by surjectivity of $\displaystyle f$, there exists an $\displaystyle x\in A$ such that $\displaystyle f(x)=y$ ...