Why is this a differentiable structure on the product manifold?

Why is this a differentiable structure on the product manifold?

pSuppose $M$ en $N$ are differentiable manifolds with differentiable
structures $\{(U_a,x_a)\}$ and $\{(V_b,x_b)\}$ resp. Consider $M\times N$
and the mappings $z_{ab}(p,q):=(x_a(p),y_b(q))$ with $p\in U_a$ and $q\in
V_b$./p pI want to prove that $\{(U_a\times V_b,z_{ab})\}$ gives a
differentiable structure on $M\times N$ such that the projections $\pi_1$
and $\pi_2$ are differentiable./p pIs this good?!:
$$\bigcup_{ab}{z_{ab}(U_a\times
V_b)}=\bigcup_a{x_a(U_a)}\times\bigcup_b{y_b(V_b)}=M\times N$$ My problem
lies in the second point of the definition of a differentiable manifold:
Suppose $z_{ab}(U_a\times V_b)\cap z_{cd}(U_c\times V_d)=W\neq\emptyset$
Why are $z^{-1}_{ab}(W)$ and $z^{-1}_{cd}(W)$ open?/p pI think
$z^{-1}_{cd}\circ z_{ab}$ is differentiable because: $$z^{-1}_{cd}\circ
z_{ab}(p,q)=z^{-1}_{cd}(z_{ab}(p,q))=((x^{-1}_c\circ
x_a)(p),(y^{-1}_d\circ y_b)(q)$$ and this is differentiable per assumption
in each component thus differentiable./p pBut why are the projections
differentiable? Herefore we have to prove that $y^{-1}_b\circ\pi\circ x_a$
are differentiable for all $a$ and $b$ right?! But how to do this? /p
pThank you for help :)/p