TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 44 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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