Theorem 5.10. Assuming that:

• $A^U$ be as above

• $J=\mathbb{N}$

• $\{C\subseteq \mathbb{N}:\mathbb{N}\setminus C\text{ is finite}\}\subseteq U$

• $n\in \mathbb{N}$

• for each $m\in \mathbb{N}$, let $B_m\subseteq A^n$ satisfying:

  • (1) $[B_m]\ne \varnothing \forall m\in \mathbb{N}$
  • (2) $[B_k]\subseteq [B_m]$ for all $m,k\in \mathbb{N}$ with $m\le k$