TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The subtraction surrogate

Statement

Let $g : B \to C$ be a morphism and let $x : X \to B$ , $y : Y \to B$ be members with $g x \equiv g y .$ Then there exists a member $z : Z \to B$ such that $g z \equiv 0.$

Moreover:

if $f : B \to D$ and $f x \equiv 0$ , then $f y \equiv f z$ ;
if $h : B \to A$ and $h y \equiv 0$ , then $h x \equiv - h z$ .

Facts & Assumptions

Given: Morphisms $g : B \to C$ , $f : B \to D$ , and $h : B \to A$ , and members $x : X \to B$ , $y : Y \to B$ with $g x \equiv g y$ .

[L1]

The relation $g x \equiv g y$ is witnessed by one common pair of epimorphisms (Equivalence of members).

[L2]

Every hom-set in an abelian category is an abelian group, so members with one common domain may be added and subtracted (Abelian category).

[L3]

Zero members and negatives behave literally under equivalence (Each object has a zero member and each member has a negative).

Proof

technique · constructive

1.1

Choose epimorphisms $u : W \to X$ and $v : W \to Y$ with $g x u = g y v$ , using [L1], and define $z : = y v - x u : W \to B$ . Then $g z = g y v - g x u = 0$ by [L2], so $g z \equiv 0$ by [L3].

L1L2L3chooseconstructalgebra

2.1

Suppose $f x \equiv 0$ . After replacing $W$ by a common epic refinement of the witnesses for $g x \equiv g y$ and $f x \equiv 0$ , we may assume $f x u = 0$ . Then $f y v = f (y v - x u) = f z$ , so the epic $v$ witnesses $f y \equiv f z$ .

L1L2L3step 1.1constructalgebra

2.2

Suppose $h y \equiv 0$ . After the same common-refinement step, we may assume $h y v = 0$ . Then $h x u = - h (y v - x u) = - h z$ , so the epic $u$ witnesses $h x \equiv - h z$ .

L1L2L3step 1.1constructalgebra

3.1

Step 1.1 gives the required member $z$ , while steps 2.1 and 2.2 prove the two moreover clauses.

step 1.1step 2.1step 2.2discharge-construct∎

Depends on

Used by

FALSE: the members of an object form an abelian group False statement
FALSE: the subtraction rule produces a unique member False statement
The cost of the member calculus Remark
Two routes to every dual statement Remark
What the subtraction rule does not say Remark

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

Saunders Mac Lane, Categories for the Working Mathematician, Theorem VIII.4.3(vi) (standard reference, not scraped)