Vector bundle class functor is homotopy-invariant for paracompact

Statement
Suppose $$A$$ and $$B$$ are paracompact Hausdorff spaces and $$f_0, f_1: A \to B$$ are continuous maps. Then, the functorially induced maps by the vector bundle class functor, namely:

$$Vect^n(f_0): Vect^n(B) \to Vect^n(A)$$

and:

$$Vect^n(f_1): Vect^n(B) \to Vect^n(A)$$

are equal, i.e. $$Vect^n(f_0) = Vect^n(f_1)$$.

Corollaries

 * Any homotopy equivalence of paracompact Hausdorff spaces induces an isomorphism on the vector bundle classes over them.
 * In particular, any vector bundle over a contractible paracompact Hausdorff space is trivial.