MATH.19:4.6 - Return the result to its use

A completed proof supplies its conclusion under the stated assumptions. Use that result in the next mathematical construction, model argument or algorithmic design. When a changed premise invalidates a lemma, reopen its dependent steps and retain arguments that still apply.

An incomplete proof plan can still locate useful work: name the remaining lemma, its available premises, and the step it would close. This can support another reader, a specialist or an AI agent without concealing the open part. If a counterexample defeats the claim, return it and the failed premise or implication.

An obstruction can also motivate changing an axiom or developing a conjecture. Carry the failed relation and the intended use into that new mathematical question.