Proposition: Applying one completion twice changes its coefficient

For a formal linear map ${\mathcal{C}}_{\kappa }=\mathrm{id}+\kappa L+O({\kappa }^2)$,

Applying the same completion again to an already completed insertion therefore requires an explicit subtraction or factorization accounting for the extra term.

Proof. Multiply the two series and compare their linear coefficients.