Proof regular languages are closed under homeomorphism
This proof looks correct. You can shorten it a bit (it doesn't need to be so descriptive), and you would get the following proof:
$h(L)=h(L(r_1\cup r_2))=h(L(r_1)\cup L(r_2)) = h(L(r_1))\cup h(L(r_2)) = L(r'_1)\cup L(r'_2)=L(r'_1\cup r'_2)$
Where the transition $h(L(r_1)\cup L(r_2)) = h(L(r_1)) \cup h(L(r_2))$ can be shown for any function $h$ (not necessarily a homeomorphism!).