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An initial universe of discourse, $A^\circ$, supplies the groundwork for any number of further extensions, beginning with the \textit{first order differential extension}, $\operatorname{E}A^\circ.$ This extends the initial alphabet, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \},$ by a \textit{first order differential alphabet}, $\operatorname{d}\mathfrak{A} = \{ ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \},$ resulting in the \textit{first order extended alphabet}, $\operatorname{E}\mathfrak{A},$ defined as follows:
An initial universe of discourse, $A^\circ$, supplies the groundwork for any number of further extensions, beginning with the \textit{first order differential extension}, $\operatorname{E}A^\circ.$ The construction of $\operatorname{E}A^\circ$ can be described in the following stages:
\begin{itemize}
\item
The initial alphabet, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \},$ is extended by a \textit{first order differential alphabet}, $\operatorname{d}\mathfrak{A} = \{ ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \},$ resulting in the \textit{first order extended alphabet}, $\operatorname{E}\mathfrak{A},$ defined as follows:
\begin{quote}
\begin{quote}
$\operatorname{E}\mathfrak{A} = \mathfrak{A} \cup \operatorname{d}\mathfrak{A} = \{ ``a_1", \ldots, ``a_n", ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \}.$
$\operatorname{E}\mathfrak{A} = \mathfrak{A}\ \cup\ \operatorname{d}\mathfrak{A} = \{ ``a_1", \ldots, ``a_n", ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \}.$
\end{quote}
\end{quote}
\item
The initial basis, $\mathcal{A} = \{ a_1, \ldots, a_n \},$ is extended by a \textit{first order differential basis}, $\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},$ resulting in the \textit{first order extended basis}, $\operatorname{E}\mathcal{A},$ defined as follows:
\begin{quote}
$\operatorname{E}\mathcal{A} = \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} = \{ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.$
\end{quote}
\end{itemize}
$\dots$
$\dots$
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==Formal development==
==Formal development==
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