MATH.23:4.5 - Revise the claim without hiding the lost use

Use the failure to choose the next mathematical change. Three common returns are:

1. Repair a hypothesis. State the condition that makes the failed lemma work, prove its sufficiency and check whether the intended cases satisfy it.
2. Repair the construction. Retain the original problem and change the objects, representation or operation so that the required property can be obtained.
3. Change the conclusion. Obtain a conditional answer, bound, obstruction or weaker property that still serves a useful question.

A sufficient condition can be worth developing before necessity is known. Label that difference: a proof that uniform convergence preserves continuity does not show that every continuous limit was obtained uniformly.

A counterexample can itself become a constructive ingredient. Ask what class it belongs to, which transformations preserve its obstruction, and which related conjectures it can test. The new question can concern the method of generating cases as well as the cases themselves.

Retain the result that survived. Changing the theorem should not erase a valid construction or an earlier unresolved requirement.