Part (a) is clear from the definitions and Algebra, Lemma \ref{algebra-lemma-standard-smooth}. Part (b) is clear from Definition \ref{definition-strictly-standard}. \medskip\noindent Proof of (c). Choose a presentation $A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$ such that (\ref{equation-elementary-standard-one}) and (\ref{equation-elementary-standard-two}) hold. Choose $h \in R[x_1, \ldots, x_n]$ mapping to $a$. Then $$ A_a = R[x_0, x_1, \ldots, x_n]/(x_0h - 1, f_1, \ldots, f_m). $$ Write $J = (x_0h - 1, f_1, \ldots, f_m)$. By (\ref{equation-elementary-standard-two}) we see that the $A_a$-module $J/J^2$ is generated by $x_0h - 1, f_1, \ldots, f_c$ over $A_a$. Hence, as in the proof of Algebra, Lemma \ref{algebra-lemma-huber}, we can choose a $g \in 1 + J$ such that $$ A_a = R[x_0, \ldots, x_n, x_{n + 1}]/ (x_0h - 1, f_1, \ldots, f_m, gx_{n + 1} - 1). $$ At this point (\ref{equation-elementary-standard-one}) implies that $R \to A_a$ is standard smooth (use the coordinates $x_0, x_1, \ldots, x_c, x_{n + 1}$ to take derivatives). \medskip\noindent Proof of (d). Choose a presentation $A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$ such that (\ref{equation-strictly-standard-one}) and (\ref{equation-strictly-standard-two}) hold. Write $I = (f_1, \ldots, f_m)$. We already know that $A_a$ is smooth over $R$, see Lemma \ref{lemma-elkik}. By Lemma \ref{lemma-parse-equation-strictly-standard-one} we see that $(I/I^2)_a$ is free on $f_1, \ldots, f_c$ and maps isomorphically to a direct summand of $\bigoplus A_a \text{d}x_i$. Since $\Omega_{A_a/R} = (\Omega_{A/R})_a$ is the cokernel of the map $(I/I^2)_a \to \bigoplus A_a \text{d}x_i$ we conclude that it is stably free. \medskip\noindent Proof of (e). Choose a presentation $A = R[x_1, \ldots, x_n]/I$ with $I$ finitely generated. By assumption we have a short exact sequence $$ 0 \to (I/I^2)_a \to \bigoplus\nolimits_{i = 1, \ldots, n} A_a\text{d}x_i \to \Omega_{A_a/R} \to 0 $$ which is split exact. Hence we see that $(I/I^2)_a \oplus \Omega_{A_a/R}$ is a free $A_a$-module. Since $\Omega_{A_a/R}$ is stably free we see that $(I/I^2)_a$ is stably free as well. Thus replacing the presentation chosen above by $A = R[x_1, \ldots, x_n, x_{n + 1}, \ldots, x_{n + r}]/J$ with $J = (I, x_{n + 1}, \ldots, x_{n + r})$ for some $r$ we get that $(J/J^2)_a$ is (finite) free. Choose $f_1, \ldots, f_c \in J$ which map to a basis of $(J/J^2)_a$. Extend this to a list of generators $f_1, \ldots, f_m \in J$. Consider the presentation $A = R[x_1, \ldots, x_{n + r}]/(f_1, \ldots, f_m)$. Then (\ref{equation-strictly-standard-two}) holds for $a^e$ for all sufficiently large $e$ by construction. Moreover, since $(J/J^2)_a \to \bigoplus\nolimits_{i = 1, \ldots, n + r} A_a\text{d}x_i$ is a split injection we can find an $A_a$-linear left inverse. Writing this left inverse in terms of the basis $f_1, \ldots, f_c$ and clearing denominators we find a linear map $\psi_0 : A^{\oplus n + r} \to A^{\oplus c}$ such that $$ A^{\oplus c} \xrightarrow{(f_1, \ldots, f_c)} J/J^2 \xrightarrow{f \mapsto \text{d}f} \bigoplus\nolimits_{i = 1, \ldots, n + r} A \text{d}x_i \xrightarrow{\psi_0} A^{\oplus c} $$ is multiplication by $a^{e_0}$ for some $e_0 \geq 1$. By Lemma \ref{lemma-parse-equation-strictly-standard-one} we see (\ref{equation-strictly-standard-one}) holds for all $a^{ce_0}$ and hence for $a^e$ for all $e$ with $e \geq ce_0$. \medskip\noindent Proof of (f). Choose a presentation $A_a = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$ such that $\det(\partial f_j/\partial x_i)_{i, j = 1, \ldots, c}$ is invertible in $A_a$. We may assume that for some $m < n$ the classes of the elements $x_1, \ldots, x_m$ correspond $a_i/1$ where $a_1, \ldots, a_m \in A$ are generators of $A$ over $R$, see Lemma \ref{lemma-standard-smooth-include-generators}. After replacing $x_i$ by $a^Nx_i$ for $m < i \leq n$ we may assume the class of $x_i$ is $a_i/1 \in A_a$ for some $a_i \in A$. Consider the ring map $$ \Psi : R[x_1, \ldots, x_n] \longrightarrow A,\quad x_i \longmapsto a_i. $$ This is a surjective ring map. By replacing $f_j$ by $a^Nf_j$ we may assume that $f_j \in R[x_1, \ldots, x_n]$ and that $\Psi(f_j) = 0$ (since after all $f_j(a_1/1, \ldots, a_n/1) = 0$ in $A_a$). Let $J = \Ker(\Psi)$. Then $A = R[x_1, \ldots, x_n]/J$ is a presentation and $f_1, \ldots, f_c \in J$ are elements such that $(J/J^2)_a$ is freely generated by $f_1, \ldots, f_c$ and such that $\det(\partial f_j/\partial x_i)_{i, j = 1, \ldots, c}$ maps to an invertible element of $A_a$. It follows that (\ref{equation-elementary-standard-one}) and (\ref{equation-elementary-standard-two}) hold for $a^e$ and all large enough $e$ as desired.