In recent years, there has been considerable research exploring connections between propositional proof systems, theories of bounded arithmetic, and complexity classes. We know that NC1 corresponds to G*0 and that P corresponds to G1 , but no proof system corresponding to a complexity class between NC1 and P has been defined.In this work, we construct a proof system GL, which corresponds to L. Connections to the theory VL (Zambella's Sp0 - rec) are also considered. GL is defined by restricting cuts in the system G1 . The first restriction is syntactic: the cut formulas have to be Sigma CNF(2), which is a new class of formulas. Unfortunately that is not enough; the free variables in cut formulas must be restricted to parameter variables. We prove that GL corresponds to VL by translating theorems of VL into tautologies with small GL proof.