Successive quotients I to n over I to n plus one assemble into a graded approximation of A. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.
The mathematical object
The \(I\)-adic filtration \(A\supset I\supset I^2\supset\cdots\) records increasing orders of vanishing. Completion replaces \(A\) by compatible residues modulo every \(I^n\).
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
One explicit computation
The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.
Why the identity matters
The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.
Where it can fail
Completion and localisation answer different questions and do not commute without hypotheses. Completeness is topological data, not merely another quotient.
The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.