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To obtain the highest confidence on the correction of numerical simulationprograms implementing the finite element method, one has to formalize themathematical notions and results that allow to establish the soundness of themethod. The Lax-Milgram theorem may be seen as one of those theoreticalcornerstones: under some completeness and coercivity assumptions, it statesexistence and uniqueness of the solution to the weak formulation of someboundary value problems. The purpose of this document is to provide the formalproof community with a very detailed pen-and-paper proof of the Lax-Milgramtheorem.

The Lax-Milgram Theorem. A detailed proof to be formalized in Coq