C. ?. Case and A. ?. If, then there is a maximal confluent set H ? rpGq with r? rp?q s H ? 0 r?s G for the application r of ? ? to A ? B ? in some ? ? for some ? P G

C. ?. Case and A. ?. Let, By monotonicity of input formulas (Lemma 5.18), A ? B ? P r?s H . By Lemma 5.14, either A ? P r?s H or B ? P r?s H . Thus, for any r?s H ? r?s G , we have by IH

C. ?. Case and 3. If, 14, there is another sequent ? P G with A ? P ? for the child ? of ? ? in tr p?q. Thus, r?s G ( A by IH. Since r?s G R 0 r?s G , we have r?s G ( 3A. Case C ? " 3A ? . Let 3A ? P r?s G and r?s G Rr?s G . Then, by Lemma 5

C. ?. Case and A. If, then there is a maximal confluent set H ? rpGq with r? rp?q s H ? r?s G for the application r of ? to A ? in some ? ? for some ? P G. In that case, A ? P r?s H for the child ? of ? rp?q in tr prp?qq. Thus, by IH r?s H * A

C. Case, A ? combines the monotonicity argument for input implications with the use of Lemma 5

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