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In the present paper, we proceed the study of framed $4$-graph minor theory initiated in ``Framed $4$-valent Graph Minor Theory I. Intoduction. Planarity Criterion '' and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in the projective plane.

Framed 4-valent graph minor theory II: Special minors and new examples