On page 114 of Milnor's book
https://aareyanmanzoor.github.io/assets/books/characteristic-classes.pdf
We have the next theorem
He says that $H^i(E,E_0,\mathbb{Z}_2)=0$ for $i<n$ as a consequence of the Thom somorphism, moreover, this fact is used in the second case of the proof. My question is about taking $j<0$ in the Thom isomorphism, since the cup product is only defined for non-negative dimensions in this book and others classical references. I think that is possible to extend the cup product for negative dimensions by definning $\phi^k\smile\psi^l=0^{k+l}$ if $k<0$ or $l<0$ in the cochain level (where $C^i=0$ for $i<0$), preserving all results about cup product (including relative Künneth formula), which would justify the assertion in theorem.
Could you confirm if this way is correct? Or do I necessarily have to use the Leray-Hirsch theorem as in Hatcher's algebraic topology book?
Thanks for the answers!