Euler class of a tensor product bundle

Question

I want to prove the result: $$c_1(L\otimes L')=c_1(L)+c_1(L')$$ where $L,L'$ are complex line bundles and $c_1(L):=e(L_{\mathbb R})$. My definition of Euler class is given by the pullback by zero section of Thom class: $$e(E):=s_0^*\Phi(E)$$ In Bott&Tu, this result is proved by writting down an explict formula of Euler class of a rank 2 bundle. I am wondering whether I can prove this by the properties of Thom class.