For, if possible, let it be in that ratio to a greater solid W;
therefore, inversely, as the base DEF is to the base ABC, so is the solid W to the pyramid ABCG.
But, as the solid W is to the solid ABCG, so is the pyramid DEFH to some solid less than the pyramid ABCG, as was before proved; [XII. 2, Lemma]
therefore also, as the base DEF is to the base ABC, so is the pyramid DEFH to some solid less than the pyramid ABCG: [V. II] which was proved absurd.
Therefore the pyramid ABCG is not to any solid greater than the pyramid DEFH as the base ABC is to the base DEF.
But it was proved that neither is it in that ratio to a less solid.
Therefore, as the base ABC is to the base DEF, so is the pyramid ABCG to the pyramid DEFH.
Q. E. D.