The proof of this lemma is made a bit longer than expected by a subtlety concerning infinite sums of rational equivalences. In the quasi-compact case the family $D_j$ is finite and the result is altogether easy and a straightforward consequence of Chow Homology, Lemma \ref{chow-lemma-compute-c1} and Divisors, Lemma \ref{divisors-lemma-c1-additive} and the definitions. \medskip\noindent The morphism $p$ is proper since the family $\{D_j\}_{j \in J}$ is locally finite. Write $\alpha = \sum_{a \in A} m_a [W_a]$ with $W_a \subset X$ an integral closed subscheme of $\delta$-dimension $k + 1$. Denote $i_a : W_a \to X$ the closed immersion. We assume that $m_a \not = 0$ for all $a \in A$ such that $\{W_a\}_{a \in A}$ is locally finite on $X$. \medskip\noindent Observe that by Chow Homology, Definition \ref{chow-definition-gysin-homomorphism} the class $i^*\alpha$ is the class of a cycle $\sum m_a\beta_a$ for certain $\beta_a \in Z_k(W_a \cap D)$. Namely, if $W_a \not \subset D$ then $\beta_a = [D \cap W_a]_k$ and if $W_a \subset D$, then $\beta_a$ is a cycle representing $c_1(\mathcal{O}_X(D)) \cap [W_a]$. \medskip\noindent For each $a \in A$ write $J = J_{a, 1} \amalg J_{a, 2} \amalg J_{a, 3}$ where \begin{enumerate} \item $j \in J_{a, 1}$ if and only if $W_a \cap D_j = \emptyset$, \item $j \in J_{a, 2}$ if and only if $W_a \not = W_a \cap D_1 \not = \emptyset$, and \item $j \in J_{a, 3}$ if and only if $W_a \subset D_j$. \end{enumerate} Since the family $\{D_j\}$ is locally finite we see that $J_{a, 3}$ is a finite set. For every $a \in A$ and $j \in J$ we choose a cycle $\beta_{a, j} \in Z_k(W_a \cap D_j)$ as follows \begin{enumerate} \item if $j \in J_{a, 1}$ we set $\beta_{a, j} = 0$, \item if $j \in J_{a, 2}$ we set $\beta_{a, j} = [D_j \cap W_a]_k$, and \item if $j \in J_{a, 3}$ we choose $\beta_{a, j} \in Z_k(W_a)$ representing $c_1(i_a^*\mathcal{O}_X(D_j)) \cap [W_j]$. \end{enumerate} We claim that $$ \beta_a \sim_{rat} \sum\nolimits_{j \in J} n_j \beta_{a, j} $$ in $\CH_k(W_a \cap D)$. \medskip\noindent Case I: $W_a \not \subset D$. In this case $J_{a, 3} = \emptyset$. Thus it suffices to show that $[D \cap W_a]_k = \sum n_j [D_j \cap W_a]_k$ as cycles. This is Lemma \ref{lemma-sum-divisors-associated-Weil}. \medskip\noindent Case II: $W_a \subset D$. In this case $\beta_a$ is a cycle representing $c_1(i_a^*\mathcal{O}_X(D)) \cap [W_a]$. Write $D = D_{a, 1} + D_{a, 2} + D_{a, 3}$ with $D_{a, s} = \sum_{j \in J_{a, s}} n_jD_j$. By Divisors, Lemma \ref{divisors-lemma-c1-additive} we have \begin{eqnarray*} c_1(i_a^*\mathcal{O}_X(D)) \cap [W_a] & = & c_1(i_a^*\mathcal{O}_X(D_{a, 1})) \cap [W_a] + c_1(i_a^*\mathcal{O}_X(D_{a, 2})) \cap [W_a] \\ & &  + c_1(i_a^*\mathcal{O}_X(D_{a, 3})) \cap [W_a]. \end{eqnarray*} It is clear that the first term of the sum is zero. Since $J_{a, 3}$ is finite we see that the last term agrees with $\sum\nolimits_{j \in J_{a, 3}} n_jc_1(i_a^*\mathcal{L}_j) \cap [W_a]$, see Divisors, Lemma \ref{divisors-lemma-c1-additive}. This is represented by $\sum_{j \in J_{a, 3}} n_j \beta_{a, j}$. Finally, by Case I we see that the middle term is represented by the cycle $\sum\nolimits_{j \in J_{a, 2}} n_j[D_j \cap W_a]_k = \sum_{j \in J_{a, 2}} n_j\beta_{a, j}$. Whence the claim in this case. \medskip\noindent At this point we are ready to finish the proof of the lemma. Namely, we have $i^*D \sim_{rat} \sum m_a\beta_a$ by our choice of $\beta_a$. For each $a$ we have $\beta_a \sim_{rat} \sum_j \beta_{a, j}$ with the rational equivalence taking place on $D \cap W_a$. Since the collection of closed subschemes $D \cap W_a$ is locally finite on $D$, we see that also $\sum m_a \beta_a \sim_{rat} \sum_{a, j} m_a\beta_{a, j}$ on $D$! (See Chow Homology, Remark \ref{chow-remark-infinite-sums-rational-equivalences}.) Ok, and now it is clear that $\sum_a m_a\beta_{a, j}$ (viewed as a cycle on $D_j$) represents $i_j^*\alpha$ and hence $\sum_{a, j} m_a\beta_{a, j}$ represents $p_* \sum_j i_j^*\alpha$ and we win.