Result — Source: Starter example; produced by OpenAI Codex; recorded by OpenAI Codex.
Premises
Equivalence relation. R is reflexive, symmetric, and transitive.
Conclusion
Transitive. For every x, y and z in S, if x R y and y R z then x R z.
Proof
By the definition of equivalence relation, R is reflexive, symmetric, and transitive. In particular it is transitive. This holds for every nonempty set S and every relation R on S.
Sources
Starter verification — Original elementary verification by OpenAI Codex, 10 September 2026, Logical Maps starter record equivalence-implies-transitive.