What do we have to show? Well, suppose we are given a collection $$ i_j : W_j \longrightarrow X $$ of closed immersions, with each $W_j$ integral of $\delta$-dimension $k + 1$ and rational functions $f_j \in R(W_j)^*$. Moreover, assume that the collection $\{i_j(W_j)\}_{j \in J}$ is locally finite on $X$. Then we have to show that $$ p_*\left(\sum i_{j, *}\text{div}(f_j)\right) $$ is rationally equivalent to zero on $X$. \medskip\noindent Note that the sum is equal to $$ \sum p_*i_{j, *}\text{div}(f_j). $$ Let $W'_j \subset Y$ be the integral closed subspace which is the image of $p \circ i_j$, see Lemma \ref{lemma-proper-image}. The collection $\{W'_j\}$ is locally finite in $Y$ by Lemma \ref{lemma-quasi-compact-locally-finite}. Hence it suffices to show, for a given $j$, that either $p_*i_{j, *}\text{div}(f_j) = 0$ or that it is equal to $i'_{j, *}\text{div}(g_j)$ for some $g_j \in R(W'_j)^*$. \medskip\noindent The arguments above therefore reduce us to the case of a since integral closed subspace $W \subset X$ of $\delta$-dimension $k + 1$. Let $f \in R(W)^*$. Let $W' = p(W)$ as above. We get a commutative diagram of morphisms $$ \xymatrix{ W \ar[r]_i \ar[d]_{p'} & X \ar[d]^p \\ W' \ar[r]^{i'} & Y } $$ Note that $p_*i_*\text{div}(f) = i'_*(p')_*\text{div}(f)$ by Lemma \ref{lemma-compose-pushforward}. As explained above we have to show that $(p')_*\text{div}(f)$ is the divisor of a rational function on $W'$ or zero. There are three cases to distinguish. \medskip\noindent The case $\dim_\delta(W') < k$. In this case automatically $(p')_*\text{div}(f) = 0$ and there is nothing to prove. \medskip\noindent The case $\dim_\delta(W') = k$. Let us show that $(p')_*\text{div}(f) = 0$ in this case. Since $(p')_*\text{div}(f)$ is a $k$-cycle, we see that $(p')_*\text{div}(f) = n[W']$ for some $n \in \mathbf{Z}$. In order to prove that $n = 0$ we may replace $W'$ by a nonempty open subspace. In particular, we may and do assume that $W'$ is a scheme. Let $\eta \in W'$ be the generic point. Let $K = \kappa(\eta) = R(W')$ be the function field. Consider the base change diagram $$ \xymatrix{ W_\eta \ar[r] \ar[d]_c & W \ar[d]^{p'} \\ \Spec(K) \ar[r]^\eta & W' } $$ Observe that $c$ is proper. Also $|W_\eta|$ has dimension $1$: use Decent Spaces, Lemma \ref{decent-spaces-lemma-topology-fibre} to identify $|W_\eta|$ as the subspace of $|W|$ of points mapping to $\eta$ and note that since $\dim_\delta(W) = k + 1$ and $\delta(\eta) = k$ points of $W_\eta$ must have $\delta$-value either $k$ or $k + 1$. Thus the local rings have dimension $\leq 1$ by Decent Spaces, Lemma \ref{decent-spaces-lemma-codimension-local-ring}. By Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-codim-1-point-in-schematic-locus} we find that $W_\eta$ is a scheme. Since $\Spec(K)$ is the limit of the nonempty affine open subschemes of $W'$ we conclude that we may assume that $W$ is a scheme by Limits of Spaces, Lemma \ref{spaces-limits-lemma-limit-is-scheme}. Then finally by the case of schemes (Chow Homology, Lemma \ref{chow-lemma-proper-pushforward-rational-equivalence}) we find that $n = 0$. \medskip\noindent The case $\dim_\delta(W') = k + 1$. In this case Lemma \ref{lemma-proper-pushforward-alteration} applies, and we see that indeed $p'_*\text{div}(f) = \text{div}(g)$ for some $g \in R(W')^*$ as desired.