P_mod_diffxxx5: fixes t :: "'a"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and f :: "'a"   and g :: "'a"   and h :: "'a"   and m :: "nat" assumes "Idomain S"   and "t \<in> carrier S"   and "t \<noteq> \<zero>\<^bsub>S\<^esub>"   and "maximal_ideal S (S \<diamondsuit>\<^sub>p t)"   and "PolynRg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y"   and "f \<in> carrier R"   and "(g, h) \<in> carrier R \<times> carrier R"   and "deg R S X (fst (g, h)) \<le> deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst (g, h)))"   and "deg R S X (snd (g, h)) + deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst (g, h))) \<le> deg R S X f"   and "0 < deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst (g, h)))"   and "0 < deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (snd (g, h)))"   and "rel_prime_pols R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst (g, h))) (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (snd (g, h)))"   and "P_mod R S X (S \<diamondsuit>\<^sub>p t) (f \<plusminus> -\<^sub>a g \<cdot>\<^sub>r h)" shows "Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m \<in> carrier R \<times> carrier R \<and> erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst (Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m)) = erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst (g, h)) \<and> erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (snd (Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m)) = erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (snd (g, h)) \<and> deg R S X (fst (Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m)) \<le> deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst (Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m))) \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc m\<^esup>)) (fst (Hpr\<^bsub> R S X t R' Y f g h\<^esub> m) \<plusminus> -\<^sub>a fst (Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m)) \<and> deg R S X (snd (Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m)) + deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst (Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m))) \<le> deg R S X f \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc m\<^esup>)) (snd (Hpr\<^bsub> R S X t R' Y f g h\<^esub> m) \<plusminus> -\<^sub>a snd (Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m)) \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc (Suc m)\<^esup>)) (f \<plusminus> -\<^sub>a fst (Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m) \<cdot>\<^sub>r snd (Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m))"