DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-29

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Sections E_x, E^y, f_x, and f^y on a product

Definition

Let $E \subseteq X \times Y$ . For $x \in X$ and $y \in Y$ , the horizontal and vertical sections of $E$ are

$E_{x} : = {y \in Y : (x, y) \in E}, E^{y} : = {x \in X : (x, y) \in E} .$

If $f : X \times Y \to Z$ is a function, its sections are

$f_{x} : Y \to Z, f_{x} (y) : = f (x, y), f^{y} : X \to Z, f^{y} (x) : = f (x, y) .$

The notation remembers which variable has been frozen: the subscript freezes $x$ , while the superscript freezes $y$ .

Depends on

Used by

Dependency tree · two levels

6 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

John K. Hunter, Measure Theory, Section 5.1 and Section 5.4 (standard reference, not scraped)