I have troubles to prove the Künneth formula for sections of vector bundles. If $E\to X,F\to Y$ are vector bundles over the algebraic varieties $X,Y$ then the formula says that $$H^0(X,E)\otimes H^0(Y,F)\cong H^0(X\times Y,E\boxtimes F)$$ where $H^0$ sets denotes the $k$-vector spaces of sections of each vector bundle, and $E\boxtimes F:=pr_X^*(E)\otimes pr_Y^*(F)$ is the tensor bundle of pullback bundles of $E,F$ by the projection coordinate maps. I've constructed a map between these spaces but I have problems to prove the map is an isomorphism. I constructed a map as follows. First, we can define a pullback map of sections, if $f:Y\to X$ and we consider a vector bundle $E\to X$ we define $$\Gamma(f):H^0(X,E)\to H^0(Y,f^*E),\qquad s\mapsto (y\mapsto\Gamma(f)(s)(y):=(y,s(f(y)))$$ which is a $k-$linear map. Using this idea we can construct the previous map taking the projections, i.e., we have $\Gamma(pr_X):H^0(X,E)\to H^0(X\times Y,pr_X^*(E))$ and $\Gamma(pr_Y):H^0(Y,F)\to H^0(X\times Y,pr_Y^*(F))$. Also, we define the map $$f:H^0(X\times Y,pr_X^*(E))\times H^0(X\times Y,pr_Y^*(F))\to H^0(X\times Y,E\boxtimes F),\quad(s,t)\mapsto s\otimes t$$
where the tensor section is constructed using the transition matrices of the vector bundles. If $g_{ij},h_{ij}$ are the transition matrices of $E,F$ respectively, then $s\in H^0(X,E),t\in H^0(Y,F)$ are given in a trivialization by regular morphisms that in the intersections verifies $s_i=g_{ij}s_j$ and $t_i=h_{ij}t_j$ and then we define a section by $$s_i\otimes t_i=(g_{ij}\otimes h_{ij})(s_j\otimes t_j)$$ Composing $f$ map and the map $$g:H^0(X,E)\times H^0(Y,F)\to H^0(X\times Y,pr_X^*(E))\times H^0(X\times Y,pr_Y^*(F))\to H^0(X\times Y,E\boxtimes F),(s,t)$$ $$(s,t)\mapsto (\Gamma(pr_X)(s),\Gamma(pr_Y)(t))$$ we obtain a bilinear map, and then we can take the unique linear map induced in the tensor product $H^0(X,E)\otimes H^0(Y,F)$. Is this map an isomorphism? And how I can prove it?