cases: fixes a1 :: "'b formula multiset"   and a2 :: "'b formula multiset"   and a3 :: "nat" assumes "a1 \<Rightarrow> a2 \<down> a3" obtains \<Gamma> :: "'b formula multiset" and \<Delta> :: "'b formula multiset" and n :: "nat" where "a1 = \<Gamma>''__" and "a2 = \<Delta>'__" and "a3 = Suc n__" and "\<bottom> \<in># \<Gamma>''__"   | k :: "'b" and \<Gamma> :: "'b formula multiset" and \<Delta> :: "'b formula multiset" and n :: "nat" where "a1 = \<Gamma>'''__" and "a2 = \<Delta>''__" and "a3 = Suc na__" and "Atom k \<in># \<Gamma>'''__" and "Atom k \<in># \<Delta>''__"   | \<Gamma> :: "'b formula multiset" and F :: "'b formula" and \<Delta> :: "'b formula multiset" and n :: "nat" where "a1 = \<^bold>\<not> F__, \<Gamma>''''__" and "a2 = \<Delta>'''__" and "a3 = Suc nb__" and "\<Gamma>''''__ \<Rightarrow> F__, \<Delta>'''__ \<down> nb__"   | F :: "'b formula" and \<Gamma> :: "'b formula multiset" and \<Delta> :: "'b formula multiset" and n :: "nat" where "a1 = \<Gamma>'''''__" and "a2 = \<^bold>\<not> Fa__, \<Delta>''''__" and "a3 = Suc nc__" and "Fa__, \<Gamma>'''''__ \<Rightarrow> \<Delta>''''__ \<down> nc__"   | F :: "'b formula" and G :: "'b formula" and \<Gamma> :: "'b formula multiset" and \<Delta> :: "'b formula multiset" and n :: "nat" where "a1 = Fb__ \<^bold>\<and> G__, \<Gamma>''''''__" and "a2 = \<Delta>'''''__" and "a3 = Suc nd__" and "Fb__, G__, \<Gamma>''''''__ \<Rightarrow> \<Delta>'''''__ \<down> nd__"   | \<Gamma> :: "'b formula multiset" and F :: "'b formula" and \<Delta> :: "'b formula multiset" and n :: "nat" and G :: "'b formula" where "a1 = \<Gamma>'''''''__" and "a2 = Fc__ \<^bold>\<and> Ga__, \<Delta>''''''__" and "a3 = Suc ne__" and "\<Gamma>'''''''__ \<Rightarrow> Fc__, \<Delta>''''''__ \<down> ne__" and "\<Gamma>'''''''__ \<Rightarrow> Ga__, \<Delta>''''''__ \<down> ne__"   | F :: "'b formula" and \<Gamma> :: "'b formula multiset" and \<Delta> :: "'b formula multiset" and n :: "nat" and G :: "'b formula" where "a1 = Fd__ \<^bold>\<or> Gb__, \<Gamma>''''''''__" and "a2 = \<Delta>'''''''__" and "a3 = Suc nf__" and "Fd__, \<Gamma>''''''''__ \<Rightarrow> \<Delta>'''''''__ \<down> nf__" and "Gb__, \<Gamma>''''''''__ \<Rightarrow> \<Delta>'''''''__ \<down> nf__"   | \<Gamma> :: "'b formula multiset" and F :: "'b formula" and G :: "'b formula" and \<Delta> :: "'b formula multiset" and n :: "nat" where "a1 = \<Gamma>'''''''''__" and "a2 = Fe__ \<^bold>\<or> Gc__, \<Delta>''''''''__" and "a3 = Suc ng__" and "\<Gamma>'''''''''__ \<Rightarrow> Fe__, Gc__, \<Delta>''''''''__ \<down> ng__"   | \<Gamma> :: "'b formula multiset" and F :: "'b formula" and \<Delta> :: "'b formula multiset" and n :: "nat" and G :: "'b formula" where "a1 = Ff__ \<^bold>\<rightarrow> Gd__, \<Gamma>''''''''''__" and "a2 = \<Delta>'''''''''__" and "a3 = Suc nh__" and "\<Gamma>''''''''''__ \<Rightarrow> Ff__, \<Delta>'''''''''__ \<down> nh__" and "Gd__, \<Gamma>''''''''''__ \<Rightarrow> \<Delta>'''''''''__ \<down> nh__"   | F :: "'b formula" and \<Gamma> :: "'b formula multiset" and G :: "'b formula" and \<Delta> :: "'b formula multiset" and n :: "nat" where "a1 = \<Gamma>" and "a2 = F \<^bold>\<rightarrow> G, \<Delta>" and "a3 = Suc n" and "F, \<Gamma> \<Rightarrow> G, \<Delta> \<down> n"