Following the propositional case, it is straightforward to prove that even in the first-order case, the forward sequent calculus is sound and complete with respect to the backward sequent calculus, and that it satisfies a strong subformula property. The completeness property is usually called ground completeness.
It is more difficult to see that working with schematic sequents and calculating most general unifiers preserves this completeness. The necessity for factoring shows that there are some subtleties. The lemma which shows that ground completeness implies completeness with schematic sequents is called a lifting lemma. We sketch the necessary definitions and proof of the lifting lemma.