The coefficientwise action is explicit

XIV-I does not state this as a separate $C^{\infty }$-linearity theorem. That bibliographic observation is true, but it does not make (eq:modulelinearity) mathematically undecidable for the displayed coefficientwise rule. The rule and action already imply it.

For an enlarged or globally glued presentation space, the action must again be specified. Conditional statements about that larger space remain conditional. The local proof does not manufacture coefficient actions on presentations which the supplied package has not enumerated.