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This paper is the second half of a two-part series devoted to the formalization of a combinatorial model of geometric varieties, generalized maps. We study here how to express continuous notions like homeomorphisms in the combinatorial world, in order to constructively describe the well-known theorem of surface classiÿcation according to genus, number of boundaries and orientability coe cient. We then use the Coq speciÿcation elaborated in the previous paper to prove the ÿrst half of this theorem, called the trading theorem.

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