CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$

For every real $x$ , $\exp (x) > 0$ and $\exp (- x) = \frac{1}{\exp (x)} .$

Given: $x \in R$ .

[L1]

[L2]

Every nonzero square in an ordered field is positive (Squares of nonzero elements are positive).

technique · direct

1.1L1algebra

Setting $y = - x$ in [L1] gives $\exp (x) \exp (- x) = 1$ , so both factors are nonzero.

2.1L1L2

Also $\exp (x) = \exp (x / 2)^{2}$ , so it is nonnegative; by step 1.1 and [L2] it is positive.

3.1step 1.1algebra∎

Dividing the identity in step 1.1 by $\exp (x)$ gives the reciprocal formula.

Depends on