TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-13

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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 integral exponential $E$ is the published exponential function

Statement

For every $x \in R$ ,

$E (x) = \exp (x) .$

Facts & Assumptions

Given: The integral exponential $E$ constructed above.

[L1]

The function $E : R \to R$ is differentiable, satisfies $E^{'} = E$ , and has $E (0) = 1$ (The inverse $E$ is differentiable, $E^{'} = E$ , and $E (0) = 1$ ).

[L2]

Every differentiable $y : R \to R$ satisfying $y^{'} = y$ and $y (0) = 1$ equals the published exponential function (The exponential is the unique solution of $y^{'} = y$ with $y (0) = 1$ ).

Proof

technique · direct

1.1

By [L1], the function $E$ is differentiable, satisfies $E^{'} = E$ , and has $E (0) = 1$ .

2.1

The uniqueness theorem [L2] therefore gives $E (x) = \exp (x)$ for every real $x$ .

step 1.1L2∎

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)

The integral exponential EE is the published exponential function