Take families $\gamma_{1},\ldots,\gamma_{n},\Gamma_{1},\ldots,\Gamma_{n}$ as above over an open neighbourhood $U$ with the two bases being Poincaré dual, i.e., $(\gamma_{i}\cap X_{b})\cdot(\Gamma_{j}\cap X_{b})=\delta_{ij}$ for $b\in U$. Let $\gamma_{1}^{*},\ldots,\gamma_{n}^{*}$ and $\Gamma_{1}^{*},\ldots,\Gamma_{n}^{*}$ be the dual bases for $\Gamma(U,R^{1}f_{*}\mathbb{Z})$ and $\Gamma(U,R^{n-1}f_{*}\mathbb{Z})$ respectively. From the choice of $\gamma_{i}$’s, we get local coordinates $y_{1},\ldots,y_{n}$ with $dy_{i}=\omega_{i}$, so in particular

$$\delta_{ij}=\omega_{i}(\partial/\partial y_{j})=\int_{\gamma_{i}\cap X_{b}}\iota(\partial/\partial y_{j})\omega,$$

hence $\iota(\partial/\partial y_{j})\omega$ defines the cohomology class $\gamma_{j}^{*}$ in $H^{1}(X_{b},\mathbb{R})$. Similarly, let

$$g_{ij}=-\int_{\Gamma_{i}\cap X_{b}}\iota(\partial/\partial y_{j})\operatorname{Im}\Omega;$$

then $-\iota(\partial/\partial y_{j})\operatorname{Im}\Omega$ defines the cohomology class $\sum_{i}g_{ij}\Gamma_{i}^{*}$ in $H^{n-1}(X_{b},\mathbb{R})$, and $\lambda_{i}=\sum_{j}g_{ij}dy_{j}$. Thus

	$\displaystyle g(\partial/\partial y_{j},\partial/\partial y_{k})$	$\displaystyle=$	$\displaystyle-\int_{X_{b}}\iota(\partial/\partial y_{j})\omega\wedge\iota(\partial/\partial y_{k})\operatorname{Im}\Omega$
		$\displaystyle=$	$\displaystyle g_{jk}.$

On the other hand, let $\check{y}_{1},\ldots,\check{y}_{n}$ be coordinates with $d\check{y}_{i}=\lambda_{i}$. Then

$${\partial\check{y}_{i}/\partial y_{j}}=g_{ij}=g_{ji}={\partial\check{y}_{j}/\partial y_{i}},$$

so $\sum\check{y}_{i}dy_{i}$ is a closed 1-form. Thus there exists locally a function $K$ such that $\partial K/\partial y_{i}=\check{y}_{i}$ and $\partial^{2}K/\partial y_{i}\partial y_{j}=g(\partial/\partial y_{i},\partial/\partial y_{j})$. A simple calculation then confirms that $\partial\check{K}/\partial\check{y}_{i}=y_{i}$. On the other hand,

	$\displaystyle g(\partial/\partial\check{y}_{i},\partial/\partial\check{y}_{j})$	$\displaystyle=$	$\displaystyle g\left(\sum_{k}{\partial y_{k}\over\partial\check{y}_{i}}{\partial\over\partial y_{k}},\sum_{\ell}{\partial y_{\ell}\over\partial\check{y}_{j}}{\partial\over\partial y_{\ell}}\right)$
		$\displaystyle=$	$\displaystyle\sum_{k,\ell}{\partial y_{k}\over\partial\check{y}_{i}}{\partial y_{\ell}\over\partial\check{y}_{j}}g(\partial/\partial y_{k},\partial/\partial y_{\ell})$
		$\displaystyle=$	$\displaystyle\sum_{k,\ell}{\partial y_{k}\over\partial\check{y}_{i}}{\partial y_{\ell}\over\partial\check{y}_{j}}{\partial\check{y}_{k}\over\partial y_{\ell}}$
		$\displaystyle=$	$\displaystyle{\partial y_{j}\over\partial\check{y}_{i}}={\partial^{2}\check{K}\over\partial\check{y}_{i}\partial\check{y}_{j}}.$

∎