A function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ has bounded variation if \begin{eqnarray} TV_{\mathbb{R}^2}(f):=\sup\limits_{\phi \in C_c^1(\mathbb{R}^2), ||\phi||_{L^{\infty}(\mathbb{R}^2)}\leq 1}\, \int\limits_{\mathbb{R}^n}f(x,y)\operatorname{div} \phi(x,y)\, dy dx < \infty. \end{eqnarray}
I am trying to prove the following characterization of the above definition of total variation of a function of two variable in terms of the total variation of the one variable function. \begin{eqnarray} TV_{\mathbb{R}^2}(f) = \int\limits_{\mathbb{R}}TV_{\mathbb{R}}(f(\cdot,y))\, dy+ \int\limits_{\mathbb{R}}TV_{\mathbb{R}}(f(x,\cdot))\, dx. \end{eqnarray}
P.S: I could prove that \begin{eqnarray} TV_{\mathbb{R}^2}(f)\leq \int\limits_{\mathbb{R}}TV_{\mathbb{R}}(f(\cdot,y))\, dy+ \int\limits_{\mathbb{R}}TV_{\mathbb{R}}(f(x,\cdot))\, dx. \end{eqnarray}
How to prove the other way round?