How to formally prove that a given hash construction is collision resistant or not?
Proof by contradiction is easy in this case.

Assume the construction is not collision resistant. Then there's an adversary who can efficiently find a pair $H(x) = H(x')$. However, that also gives them $H_1(x) = H_1(x')$ and $H_2(x) = H_2(x')$, so neither hash function is collision resistant, which contradicts the assumption.