LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Half-space extensions agreeing on a relatively open set have the same derivatives there

Statement

If two smooth Euclidean extensions agree on a relatively open subset of $H^{n}$ , then all of their derivatives agree at every point of that subset.

Facts & Assumptions

Given: A relatively open set $U \subseteq H^{n}$ and two smooth Euclidean maps $F$ and $G$ , defined on neighbourhoods of $U$ , whose restrictions to $U$ agree.

Proof

technique · direct

1.1given

Their difference vanishes on the relative interior, which is dense in the relative set. Every Euclidean derivative of the difference vanishes there by ordinary differentiation.

2.1step 1.1∎

Those derivatives are continuous, so they also vanish at each face point. Hence the derivative of a half-space-smooth map is independent of its chosen extension.

Depends on

Smooth functions on relatively open half-space sets

Half-space extensions agreeing on a relatively open set have the same derivatives there

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources