The extensions of sequent calculi by rules, presented in the previous chapter, share the good structural properties of the purely logical G3-calculi, i.e., the rules of weakening, contraction, and cut are admissible. In addition to being admissible, weakening and contraction are height-preserving admissible. The usual consequence of cut elimination, the subformula property, holds in a weaker form, because all the formulas in the derivations in such extensions are subformulas of the endsequent or atomic formulas. However, by analysing, analogously to natural deduction, minimal derivations in specific theories, we can establish a subterm property, by which all terms in a derivation can be restricted to terms in the endsequent.

This chapter gives proofs of the subterm property for partial and linear order, the latter not an easy result. To make its presentation manageable, a system of rules that act on the right part of multisuccedent sequents is used. Further, it is shown through a proof-theoretical algorithm how to linearize a partial order, a result known as Szpilrajn's theorem. The extension is based on the conservativity of the rule system for linear order over that for partial order for sequents that have just one atom in the succedent. Finally, the proof-theoretical solution of the word problem for lattices of Chapter 4 is extended to linear lattices, i.e., lattices in which the order relation is linear.