Consider the triangle constructed in Lemma \ref{lemma-distinguished-triangle-blowup}. We claim that the map $$ Rb_*(\Omega^\bullet_{X'/S}) \oplus i_*\Omega^\bullet_{Z/S} \to i_*Rp_*(\Omega^\bullet_{E/S}) $$ has a splitting whose image contains the summand $i_*\Omega^\bullet_{Z/S}$. By Derived Categories, Lemma \ref{derived-lemma-split} this will show that the first arrow of the triangle has a splitting which vanishes on the summand $i_*\Omega^\bullet_{Z/S}$ which proves the lemma. We will prove the claim by decomposing $Rp_*\Omega^\bullet_{E/S}$ into a direct sum where the first piece corresponds to $\Omega^\bullet_{Z/S}$ and the second piece can be lifted through $Rb_*\Omega^\bullet_{X'/S}$. \medskip\noindent Proof of the claim. We may decompose $X$ into open and closed subschemes having fixed relative dimension to $S$, see Morphisms, Lemma \ref{morphisms-lemma-smooth-omega-finite-locally-free}. Since the derived category $D(X, f^{-1}\mathcal{O})_S)$ correspondingly decomposes as a product of categories, we may assume $X$ has fixed relative dimension $N$ over $S$. We may decompose $Z = \coprod Z_m$ into open and closed subschemes of relative dimension $m \geq 0$ over $S$. The restriction $i_m : Z_m \to X$ of $i$ to $Z_m$ is a regular immersion of codimension $N - m$, see Divisors, Lemma \ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion}. Let $E = \coprod E_m$ be the corresponding decomposition, i.e., we set $E_m = p^{-1}(Z_m)$. If $p_m : E_m \to Z_m$ denotes the restriction of $p$ to $E_m$, then we have a canonical isomorphism $$ \tilde \xi_m : \bigoplus\nolimits_{t = 0, \ldots, N - m - 1} \Omega^\bullet_{Z_m/S}[-2t] \longrightarrow Rp_{m, *}\Omega^\bullet_{E_m/S} $$ in $D(Z_m, (Z_m \to S)^{-1}\mathcal{O}_S)$ where in degree $0$ we have the canonical map $\Omega^\bullet_{Z_m/S} \to Rp_{m, *}\Omega^\bullet_{E_m/S}$. See Remark \ref{remark-projective-space-bundle-formula}. Thus we have an isomorphism $$ \tilde \xi : \bigoplus\nolimits_m \bigoplus\nolimits_{t = 0, \ldots, N - m - 1} \Omega^\bullet_{Z_m/S}[-2t] \longrightarrow Rp_*(\Omega^\bullet_{E/S}) $$ in $D(Z, (Z \to S)^{-1}\mathcal{O}_S)$ whose restriction to the summand $\Omega^\bullet_{Z/S} = \bigoplus \Omega^\bullet_{Z_m/S}$ of the source is the canonical map $\Omega^\bullet_{Z/S} \to Rp_*(\Omega^\bullet_{E/S})$. Consider the subcomplexes $M_m$ and $K_m$ of the complex $\bigoplus\nolimits_{t = 0, \ldots, N - m - 1} \Omega^\bullet_{Z_m/S}[-2t]$ introduced in Remark \ref{remark-projective-space-bundle-formula}. We set $$ M = \bigoplus M_m \quad\text{and}\quad K = \bigoplus K_m $$ We have $M = K[-2]$ and by construction the map $$ c_{E/Z} \oplus \tilde \xi|_M : \Omega^\bullet_{Z/S} \oplus M \longrightarrow Rp_*(\Omega^\bullet_{E/S}) $$ is an isomorphism (see remark referenced above). \medskip\noindent Consider the map $$ \delta : \Omega^\bullet_{E/S}[-2] \longrightarrow \Omega^\bullet_{X'/S} $$ in $D(X', (X' \to S)^{-1}\mathcal{O}_S)$ of Lemma \ref{lemma-gysin-via-log-complex} with the property that the composition $$ \Omega^\bullet_{E/S}[-2] \longrightarrow \Omega^\bullet_{X'/S} \longrightarrow \Omega^\bullet_{E/S} $$ is the map $\theta'$ of Remark \ref{remark-cup-product-as-a-map} for $c_1^{dR}(\mathcal{O}_{X'}(-E))|_E) = c_1^{dR}(\mathcal{O}_E(1))$. The final assertion of Remark \ref{remark-projective-space-bundle-formula} tells us that the diagram $$ \xymatrix{ K[-2] \ar[d]_{(\tilde \xi|_K)[-2]} \ar[r]_{\text{id}} & M \ar[d]^{\tilde x|_M} \\ Rp_*\Omega^\bullet_{E/S}[-2] \ar[r]^-{Rp_*\theta'} & Rp_*\Omega^\bullet_{E/S} } $$ commutes. Thus we see that we can obtain the desired splitting of the claim as the map \begin{align*} Rp_*(\Omega^\bullet_{E/S}) & \xrightarrow{(c_{E/Z} \oplus \tilde \xi|_M)^{-1}} \Omega^\bullet_{Z/S} \oplus M \\ & \xrightarrow{\text{id} \oplus \text{id}^{-1}} \Omega^\bullet_{Z/S} \oplus K[-2] \\ & \xrightarrow{\text{id} \oplus (\tilde \xi|_K)[-2]} \Omega^\bullet_{Z/S} \oplus Rp_*\Omega^\bullet_{E/S}[-2] \\ & \xrightarrow{\text{id} \oplus Rb_*\delta} \Omega^\bullet_{Z/S} \oplus Rb_*\Omega^\bullet_{X'/S} \end{align*} The relationship between $\theta'$ and $\delta$ stated above together with the commutative diagram involving $\theta'$, $\tilde \xi|_K$, and $\tilde \xi|_M$ above are exactly what's needed to show that this is a section to the canonical map $\Omega^\bullet_{Z/S} \oplus Rb_*(\Omega^\bullet_{X'/S}) \to Rp_*(\Omega^\bullet_{E/S})$ and the proof of the claim is complete.