Observe that $f^*\mathcal{A} = \colim f^*\mathcal{A}_i$ and that $$ f^*\mathcal{A}_i = \mathcal{O}'\langle f^{-1}\mathcal{S}_0 \amalg \ldots \amalg f^{-1}\mathcal{S}_i\rangle $$ with differential given by the inductive procedure above using $f^{-1}\delta_{i + 1}$. Thus it suffices to prove that $\mathcal{A}$ is flat as a graded $\mathcal{O}$-module and is K-flat as a complex of $\mathcal{O}$-modules. For this it suffices to prove that each $\mathcal{A}_i$ is flat as a graded $\mathcal{O}$-module and is K-flat as a complex of $\mathcal{O}$-modules, compare with Lemma \ref{lemma-good-direct-sum}. \medskip\noindent For $i \geq 1$ write $\mathcal{S} = \mathcal{S}_0 \amalg \ldots \amalg \mathcal{S}_i$ so that we have $\mathcal{A}_i = \mathcal{O}\langle \mathcal{S} \rangle$ as a graded $\mathcal{O}$-algebra. We are going to construct a filtration of this algebra by differential graded $\mathcal{O}$-submodules. \medskip\noindent Set $W = \mathbf{Z}_{\geq 0}^{i + 1}$ considered with lexicographical ordering. Namely, given $w = (w_0, \ldots w_i)$ and $w' = (w'_0, \ldots, w'_i)$ in $W$ we say $$ w > w' \Leftrightarrow \exists j,\ 0 \leq j \leq i : w_i = w'_i,\ w_{i - 1} = w'_{i - 1},\ \ldots , \ w_{j + 1} = w'_{j + 1},\ w_j > w'_j $$ and so on. Suppose given a section $s = s_1 \cdot \ldots \cdot s_r$ of $\mathcal{S} \times \ldots \times \mathcal{S}$ over $U$. We say that the {\it weight of $s$ is defined} if we have $s_a \in \mathcal{S}_{j_a}(U)$ for a unique $0 \leq j_a \leq i$. In this case we define the weight $$ w(s) = (w_0(s), \ldots, w_i(s)) \in W,\quad w_j(s) = |\{a \mid j_a = j\}| $$ The weight of any section of $\mathcal{S} \times \ldots \times \mathcal{S}$ is defined locally. The reader checks easily that we obtain a disjoint union decompostion $$ \mathcal{S} \times \ldots \times \mathcal{S} = \coprod\nolimits_{w \in W} \left( \mathcal{S} \times \ldots \times \mathcal{S}\right)_w $$ into the subsheaves of sections of a given weight. Of course only $w \in W$ with $\sum_{0 \leq j \leq i} w_j = r$ show up for a given $r$. We correspondingly obtain a decomposition $$ \mathcal{A}_i = \mathcal{O} \oplus \bigoplus\nolimits_{r \geq 1} \bigoplus\nolimits_{w \in W} \mathcal{O}[\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_w] $$ The rest of the proof relies on the following trivial observation: given $r$, $w$ and local section $s = s_1 \cdot \ldots \cdot s_r$ of $\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_w$ we have $$ \text{d}(s) \text{ is a local section of } \mathcal{O} \oplus \bigoplus\nolimits_{r' \geq 1} \bigoplus\nolimits_{w' \in W,\ w' < w} \mathcal{O}[\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_{w'}] $$ The reason is that in each of the expressions $$ (-1)^{\deg(s_1) + \ldots + \deg(s_{a - 1})} s_1 \cdot \ldots s_{a - 1} \cdot \delta(s_a) \cdot s_{a + 1} \cdot \ldots \cdot s_r $$ whose sum give the element $\text{d}(s)$ the element $\delta(s_a)$ is locally a $\mathcal{O}$-linear combination of elements $s'_1 \cdot \ldots \cdot s'_{r'}$ with $s'_{a'}$ in $\mathcal{S}_{j'_a}$ for some $0 \leq j'_{a'} < j_a$ where $j_a$ is such that $s_a$ is section of $\mathcal{S}_{j_a}$. \medskip\noindent What this means is the following. Suppose for $w \in W$ we set $$ F_w \mathcal{A}_i = \mathcal{O} \oplus \bigoplus\nolimits_{r \geq 1} \bigoplus\nolimits_{w' \in W,\ w' \leq w} \mathcal{O}[\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_{w'}] $$ By the observation above this is a differential graded $\mathcal{O}$-submodule. We get admissible short exact sequences $$ 0 \to \colim_{w' < w} F_{w'}\mathcal{A}_i \to F_w\mathcal{A}_i \to \bigoplus\nolimits_{r \geq 1} \mathcal{O}[\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_w] \to 0 $$ of differential graded $\mathcal{A}$-modules where the differential on the right hand side is zero. \medskip\noindent Now we finish the proof by transfinite induction over the ordered set $W$. The differential graded complex $F_0\mathcal{A}_0$ is the summand $\mathcal{O}$ and this is K-flat and graded flat. For $w \in W$ if the result is true for $F_{w'}\mathcal{A}_i$ for $w' < w$, then by Lemmas \ref{lemma-good-direct-sum}, \ref{lemma-good-admissible-ses}, and \ref{lemma-free-graded-module-good} we obtain the result for $w$. Finally, we have $\mathcal{A}_i = \colim_{w \in W} F_w\mathcal{A}_i$ and we conclude.