If $f:X_{1}\rightarrow X_{2}$ is a morphism of varieties over $K$, then we define the push-forward of $X_{1}$ with respect to $f$ as the cycle $f_{*}(X_{1}):=\deg(f)\overline{f(X_{1})}$, where the degree of $f$ is defined as $\deg(f):=[K(X_{1}):K(f(X_{1}))]$ if $f$ is generically finite and we set $\deg(f):=0$ if $[K(X_{1}):K(f(X_{1}))]=\infty$. By restriction, the push-forward can be defined in the same way on prime cycles of $X_{1}$ and extends by linearity to all cycles of $X_{1}$.

Now let $\varphi:U\rightarrow T={\mathbb{G}}_{m}^{r}$ be a moment map of the open subset $U$ of $X$. By 4.6,

is a tropical cycle on ${\mathbb{R}}^{r}$. If $\varphi$ is generically finite, then this tropical cycle is of pure dimension $\dim(X)$ and the support is equal to ${\varphi_{\rm trop}}({U^{\rm an}})$ (see Lemma 4.9).