by (metis \<open>N \<subseteq> M\<close> \<open>infinite N\<close> less_Suc_eq)

next

case False

then have "i = r-1"

...

...

@@ -822,22 +800,20 @@ proof (induction r)

using fim i False \<open>i<r\<close> by (force simp: h_def)

then obtain P j where "P \<subseteq> N" "infinite P" "j<r" and j: "\<forall>k<r. j\<noteq>k \<longrightarrow> h -` {k} \<inter> \<F> \<inter> Pow P = {}"

by (metis Ramsey_def Ram \<open>infinite N\<close>)

show ?thesis

have "\<exists>i. \<forall>j<Suc r. j \<noteq> i \<longrightarrow> f -` {j} \<inter> \<F> \<inter> Pow P = {}"