cotangent bundle is a bundle

Verifying the first criterion is simply a matter of writing it out:

(x^{1},\ldots,x^{2n})\mapsto(\pi(x^{1},\ldots,x^{2n}),\phi_{\alpha}(x^{1},% \ldots,x^{n}))=((x^{1},\ldots,x^{n}),(x^{n+1},\ldots,x^{2n}))

This is obviously a homeomorphism.

As for the second criterion,

$\displaystyle\sum_{j=1}^{n}{g_{\alpha\beta}}^{i}_{j}(x^{1},\ldots,x^{2n}){\phi% _{\beta}}^{j}(x^{1},\ldots,x^{2n})$	$\displaystyle=$	$\displaystyle\sum_{j=n}^{2n}{\partial\big{(}\sigma_{\alpha\beta}(x^{1},\ldots x% ^{n})\big{)}^{i}\over\partial x^{j}}x^{j+n}$
	$\displaystyle=$	$\displaystyle{{\sigma^{\prime}}_{\alpha\beta}}^{i+n}(x^{1},\ldots,x^{2n})$
	$\displaystyle=$	$\displaystyle{\phi_{\alpha}}^{i}(x^{1},\ldots,x^{2n})$

The third criterion follows from the chain rule:

\displaystyle{g_{\alpha\beta}}^{i}_{j}{g_{\beta\gamma}}^{j}_{k}={\partial\big{% (}\sigma_{\alpha\beta}(x^{1},\ldots x^{n})\big{)}^{i}\over\partial x^{j}}{% \partial\big{(}\sigma_{\beta\gamma}(x^{1},\ldots x^{n})\big{)}^{j}\over% \partial x^{k}}