MATH.22:4.1 - Identify the assumption and its mathematical job

State the objects, operations and laws currently used. Find the step where the disputed assumption enters. It may justify rearranging operations, comparing two objects, choosing an element, taking a limit, or introducing an object with a required property.

Distinguish a law about the objects from a logical rule used to infer statements about them. Removing commutativity changes the structures being studied. Restricting proof by contradiction can change the permitted arguments even when the question concerns the same objects.

State the desired gain. Examples include admitting noncommuting operations, retaining incomparable alternatives, obtaining a missing limit, or preserving the computational content of a proof. This gain guides which consequences to examine first.

For a local question, the relevant definitions and proof dependencies are enough. Reconstruct a whole formal foundation only when the conclusion being sought depends on it.