For any $x \in Z$ we can choose an affine open neighbourhood $\Spec(A) \subset X$ such that $Z \cap \Spec(A) = V(f_1, \ldots, f_r)$ where $f_1, \ldots, f_r \in A$ is a regular sequence. See Divisors, Definition \ref{divisors-definition-regular-immersion} and Lemma \ref{divisors-lemma-Noetherian-scheme-regular-ideal}. Then we see that the Koszul complex on $f_1, \ldots, f_r$ is a resolution of $A/(f_1, \ldots, f_r)$ for example by More on Algebra, Lemma \ref{more-algebra-lemma-regular-koszul-regular}. Hence $A/(f_1, \ldots, f_r)$ is perfect as an $A$-module. It follows that $F = i_*\mathcal{O}_Z$ is a perfect object of $D(\mathcal{O}_X)$ whose restriction to $X \setminus Z$ is zero. Since $X$ is quasi-compact and quasi-separated (Properties, Lemma \ref{properties-lemma-locally-Noetherian-quasi-separated}) and has the resolution property we see that $F = i_*\mathcal{O}_Z$ can be represented by a bounded complex of finite locally free modules (Derived Categories of Schemes, Lemma \ref{perfect-lemma-construct-strictly-perfect}) and hence the Chern classes of $F = i_*\mathcal{O}_Z$ are defined (Definition \ref{definition-defined-on-perfect}). All in all we conclude that the statement makes sense. \medskip\noindent Denote $b : W \to \mathbf{P}^1_X$ the blowing up in $\infty(Z)$ as in Section \ref{section-blowup-Z-first}. By (\ref{item-find-Z-in-blowup}) we have a closed immersion $$ i' : \mathbf{P}^1_Z \longrightarrow W $$ We claim that $Q = i'_*\mathcal{O}_{\mathbf{P}^1_Z}$ is a perfect object of $D(\mathcal{O}_W)$ and that $F$ and $Q$ satisfy the assumptions of Lemma \ref{lemma-independent-loc-chern-bQ}. \medskip\noindent Assume the claim. The output of Lemma \ref{lemma-independent-loc-chern-bQ} is that we have $$ c_p(Z \to X, F) = c'_p(Q) = (E \to Z)_* \circ c'_p(Q|_E) \circ C $$ for all $p \geq 1$. Observe that $Q|_E$ is equal to the pushforward of the structure sheaf of $Z$ via the morphism $Z \to E$ which is the base change of $i'$ by $\infty$. Thus the vanishing of $c_t(Z \to X, F)$ for $1 \leq t \leq r - 1$ by Lemma \ref{lemma-compute-section} applied to $E \to Z$. Because $\mathcal{C}_{Z/X} = \mathcal{N}^\vee$ is locally free the bivariant class $c(Z \to X, \mathcal{N})$ is characterized by the relation $$ j^* \circ C = p^* \circ c(Z \to X, \mathcal{N}) $$ where $j : C_ZX \to W_\infty$ and $p : C_ZX \to Z$ are the given maps. (Recall $C \in A^0(W_\infty \to X)$ is the class of Lemma \ref{lemma-gysin-at-infty}.) Thus the displayed equation in the statement of the lemma follows from the corresponding equation in Lemma \ref{lemma-compute-section}. \medskip\noindent Proof of the claim. Let $A$ and $f_1, \ldots, f_r$ be as above. Consider the affine open $\Spec(A[s]) \subset \mathbf{P}^1_X$ as in Section \ref{section-blowup-Z-first}. Recall that $s = 0$ defines $(\mathbf{P}^1_X)_\infty$ over this open. Hence over $\Spec(A[s])$ we are blowing up in the ideal generated by the regular sequence $s, f_1, \ldots, f_r$. By More on Algebra, Lemma \ref{more-algebra-lemma-blowup-regular-sequence} the $r + 1$ affine charts are global complete intersections over $A[s]$. The chart corresponding to the affine blowup algebra $$ A[s][f_1/s, \ldots, f_r/s] = A[s, y_1, \ldots, y_r]/(sy_i - f_i) $$ contains $i'(Z \cap \Spec(A))$ as the closed subscheme cut out by $y_1, \ldots, y_r$. Since $y_1, \ldots, y_r, sy_1 - f_1, \ldots, sy_r - f_r$ is a regular sequence in the polynomial ring $A[s, y_1, \ldots, y_r]$ we find that $i'$ is a regular immersion. Some details omitted. As above we conclude that $Q = i'_*\mathcal{O}_{\mathbf{P}^1_Z}$ is a perfect object of $D(\mathcal{O}_W)$. Since $W$ also has the resolution property (Derived Categories of Schemes, Lemma \ref{perfect-lemma-resolution-property-ample-relative}) we find that the Chern classes of $Q$ are defined. All the other assumptions on $F$ and $Q$ in Lemma \ref{lemma-independent-loc-chern-bQ} (and Lemma \ref{lemma-localized-chern-pre}) are immediately verified.