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:* <p>Finally, the initial universe, <math>A^\circ = [ a_1, \ldots, a_n ],</math> is extended by a ''first order differential universe'' or ''tangent universe'', <math>\operatorname{d}A^\circ = [ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ],</math> at each point of <math>A^\circ,</math> resulting in a ''first order extended universe'' or ''tangent bundle universe'', <math>\operatorname{E}A^\circ,</math> defined as follows:</p><blockquote><math>\operatorname{E}A^\circ = [ \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 ].</math></blockquote><p>This gives <math>\operatorname{E}A^\circ</math> the type:</p><blockquote><math>[ \mathbb{B}^n \times \mathbb{D}^n ] = (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).</math></blockquote>

:* <p>Finally, the initial universe, <math>A^\circ = [ a_1, \ldots, a_n ],</math> is extended by a ''first order differential universe'' or ''tangent universe'', <math>\operatorname{d}A^\circ = [ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ],</math> at each point of <math>A^\circ,</math> resulting in a ''first order extended universe'' or ''tangent bundle universe'', <math>\operatorname{E}A^\circ,</math> defined as follows:</p><blockquote><math>\operatorname{E}A^\circ = [ \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 ].</math></blockquote><p>This gives <math>\operatorname{E}A^\circ</math> the type:</p><blockquote><math>[ \mathbb{B}^n \times \mathbb{D}^n ] = (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).</math></blockquote>

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A proposition in a differential extension of a universe of discourse is called a ''differential proposition'' and forms the analogue of a system of differential equations in ordinary calculus.  With these constructions, the first order extended universe <math>\operatorname{E}A^\circ</math> and the first order differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at the foothills of [[differential logic]].

Table 5 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a corresponding manner.

Table 5 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.

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