finitely generated Module over a local ring and fibres of a vector space basis as generators

Let be a local ring with Residue field

Suppose is a finitely generated -modules, the vector space over (cf. functor from modules to vector spaces for the residue field).
Assume further that generate as module, and are fibres of .

Then form a generating set

w.l.o.g. we may assume that form a -basis

Let be the generated submodule .
Then it remains to show that

consider the sequence of modules

it follows especially that the composition

is surjective, as form a k-basis of and hence especially a generating set over

using the first isomorphism theorem for modules there exists a diagram

where

and hence an isomorphism

This shows

by second Nakaymas Lemma for Jacobson Radical we conclude that