Fix $k \in \mathbf{Z}$. We first show the map is injective. Suppose that $(\alpha_0, \ldots, \alpha_{r - 1})$ is an element of the left hand side that maps to zero. By Lemma \ref{lemma-cap-projective-bundle} we see that $$ 0 = \pi_*(\pi^*\alpha_0 + c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}_P(1))^{r - 1} \cap \pi^*\alpha_{r-1}) = \alpha_{r - 1} $$ Next, we see that $$ 0 = \pi_*(c_1(\mathcal{O}_P(1)) \cap (\pi^*\alpha_0 + c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}_P(1))^{r - 2} \cap \pi^*\alpha_{r - 2})) = \alpha_{r - 2} $$ and so on. Hence the map is injective. \medskip\noindent It remains to show the map is surjective. Let $X_i$, $i \in I$ be the irreducible components of $X$. Then $P_i = \mathbf{P}(\mathcal{E}|_{X_i})$, $i \in I$ are the irreducible components of $P$. Consider the commutative diagram $$ \xymatrix{ \coprod P_i \ar[d]_{\coprod \pi_i} \ar[r]_p & P \ar[d]^\pi \\ \coprod X_i \ar[r]^q & X } $$ Observe that $p_*$ is surjective. If $\beta \in \CH_k(\coprod X_i)$ then $\pi^* q_* \beta = p_*(\coprod \pi_i)^* \beta$, see Lemma \ref{lemma-flat-pullback-proper-pushforward}. Similarly for capping with $c_1(\mathcal{O}(1))$ by Lemma \ref{lemma-pushforward-cap-c1}. Hence, if the map of the lemma is surjective for each of the morphisms $\pi_i : P_i \to X_i$, then the map is surjective for $\pi : P \to X$. Hence we may assume $X$ is irreducible. Thus $\dim_\delta(X) < \infty$ and in particular we may use induction on $\dim_\delta(X)$. \medskip\noindent The result is clear if $\dim_\delta(X) < k$. Let $\alpha \in \CH_{k + r - 1}(P)$. For any locally closed subscheme $T \subset X$ denote $\gamma_T : \bigoplus \CH_{k + i}(T) \to \CH_{k + r - 1}(\pi^{-1}(T))$ the map $$ \gamma_T(\alpha_0, \ldots, \alpha_{r - 1}) = \pi^*\alpha_0 + \ldots + c_1(\mathcal{O}_{\pi^{-1}(T)}(1))^{r - 1} \cap \pi^*\alpha_{r - 1}. $$ Suppose for some nonempty open $U \subset X$ we have $\alpha|_{\pi^{-1}(U)} = \gamma_U(\alpha_0, \ldots, \alpha_{r - 1})$. Then we may choose lifts $\alpha'_i \in \CH_{k + i}(X)$ and we see that $\alpha - \gamma_X(\alpha'_0, \ldots, \alpha'_{r - 1})$ is by Lemma \ref{lemma-restrict-to-open} rationally equivalent to a $k$-cycle on $P_Y = \mathbf{P}(\mathcal{E}|_Y)$ where $Y = X \setminus U$ as a reduced closed subscheme. Note that $\dim_\delta(Y) < \dim_\delta(X)$. By induction the result holds for $P_Y \to Y$ and hence the result holds for $\alpha$. Hence we may replace $X$ by any nonempty open of $X$. \medskip\noindent In particular we may assume that $\mathcal{E} \cong \mathcal{O}_X^{\oplus r}$. In this case $\mathbf{P}(\mathcal{E}) = X \times \mathbf{P}^{r - 1}$. Let us use the stratification $$ \mathbf{P}^{r - 1} = \mathbf{A}^{r - 1} \amalg \mathbf{A}^{r - 2} \amalg \ldots \amalg \mathbf{A}^0 $$ The closure of each stratum is a $\mathbf{P}^{r - 1 - i}$ which is a representative of $c_1(\mathcal{O}(1))^i \cap [\mathbf{P}^{r - 1}]$. Hence $P$ has a similar stratification $$ P = U^{r - 1} \amalg U^{r - 2} \amalg \ldots \amalg U^0 $$ Let $P^i$ be the closure of $U^i$. Let $\pi^i : P^i \to X$ be the restriction of $\pi$ to $P^i$. Let $\alpha \in \CH_{k + r - 1}(P)$. By Lemma \ref{lemma-pullback-affine-fibres-surjective} we can write $\alpha|_{U^{r - 1}} = \pi^*\alpha_0|_{U^{r - 1}}$ for some $\alpha_0 \in \CH_k(X)$. Hence the difference $\alpha - \pi^*\alpha_0$ is the image of some $\alpha' \in \CH_{k + r - 1}(P^{r - 2})$. By Lemma \ref{lemma-pullback-affine-fibres-surjective} again we can write $\alpha'|_{U^{r - 2}} = (\pi^{r - 2})^*\alpha_1|_{U^{r - 2}}$ for some $\alpha_1 \in \CH_{k + 1}(X)$. By Lemma \ref{lemma-relative-effective-cartier} we see that the image of $(\pi^{r - 2})^*\alpha_1$ represents $c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1$. We also see that $\alpha - \pi^*\alpha_0 - c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1$ is the image of some $\alpha'' \in \CH_{k + r - 1}(P^{r - 3})$. And so on.