Adapting a SAT resolution proof into a Max-SAT resolution proof without considerably increasing its size is an open problem. Read-once resolution, where each clause is used at most once in the proof, represents the only fragment of resolution for which an adaptation using exclusively Max-SAT resolution is known and trivial. Proofs containing non read-once clauses are difficult to adapt because the Max-SAT resolution rule replaces the premises by the conclusions. This paper contributes to this open problem by defining, for the first time since the introduction of Max-SAT resolution, a new fragment of resolution whose proofs can be adapted to Max-SAT resolution proofs without substantially increasing their size. In this fragment, called crossing-free resolution, non read-once clauses are used independently to infer new information thus enabling to bring along each non read-once clause while unfolding the proof until a substitute is required.