A weakened version of the Cutting Plane (CP) proof system with a restriction on the degree of falsity of intermediate inequalities was introduced by Goerdt. He proved an exponential lower bound for CP proofs with degree of falsity bounded by n/log²n+1 where n is the number of variables. Hirsch and Nikolenko strengthened this result by establishing a direct connection between CP and Resolution proofs. This result implies an exponential lower bound on the proof length of the Tseitin-Urquhart tautologies, when the degree of falsity is bounded by cn for some constant c. In this paper we generalize this result for extensions of Lovasz-Schrijver calculi (LS), namely for LS^k+CP^k proof systems introduced by Grigoriev et al. We show that any LS^k+CP^k proof with bounded degree of falsity can be transformed into a Res(k) proof. We also prove lower and upper bounds for the new system.