We give a formal and constructive proof in Coq/Ssreflect of the known result that the graphs of treewidth two are exactly those that do not admit K4 as a minor. This result is a milestone towards a formal proof of the recent result that isomorphism of treewidth-two graphs can be finitely axiomatized. The proof is based on a function extracting terms from K4-free graphs in such a way that the interpretation of an extracted term yields a treewidth-two graph isomorphic to the original graph.