# Show these two axioms of Stiefel-Whitney class are equivalent.

Here is a theorem about Stiefel-Whitney class in my teacher's notes:

For a dim-$$k$$ real vector bundles $$E$$ over $$B, k \geq 0,$$ there are characteristic classes $$w_{i}(E) \in H^{i}(B, \mathbb{Z}/2\mathbb{Z}),$$ called the Stiefel-Whitney classes. They satisfy and are uniquely determined by the following axioms:

1. $$w_{0}(E)=1, w_{i}(E)=0$$ for $$i>k$$.
2. $$w_{1}(L)$$ is the mod $$2$$ Euler class $$e_{2}(L)$$ for any real line bundle.
3. $$f^{*} w_{i}(E)=w_{i}\left(f^{*} E\right)$$.
4. $$w\left(E_{1} \oplus E_{2}\right)=w\left(E_{1}\right) \cup w\left(E_{2}\right)$$ or equivalently $$w_{i}\left(E_{1} \oplus E_{2}\right)=\sum_{j=0}^{j=i} w_{i-j}\left(E_{1}\right) \cup w_{j}\left(E_{2}\right)$$.

However, in many other references, like wikipedia, the second axiom is replaced by

Normalization: The Whitney class of the tautological line bundle over the real projective space $$\mathbf {P}^1(\mathbf{R})$$ is nontrivial, i.e. $${w(\gamma _{1}^{1})=1+a\in H^{*}(\mathbf {P} ^{1}(\mathbf{R});\mathbf {Z} /2\mathbf{Z} )=(\mathbf {Z} /2\mathbf {Z} )[a]/(a^{2})}$$.

Could you please tell me how to show that these two axioms are equivalent? Thank you for your help!