are κ-partitions of 20 n .

Take the disjoint union 20 (with 20 = (W, F20, R20)) of the 20n, –1 ≤ n ≤ m, and 20g.

Define A = (A1, A2) in M by putting

$${\mathcal{A}}_{1}(x)={\mathcal{A}}_{1}^{a}(x)\cup\bigcup_{-1\leq i\leq m}{\mathcal{A}}_{1}^{i}(x),$$

for all set variables x and A 2 ( b ) = A 2 ( b ), for all object variables b . Let, for 0 < n < m,

$$X_{n}\,=\,Z_{n}\,\cup\,\bigcup_{n\,\leq i\,\leq m}W_{i}\,.$$

We show that (20, A) and the sets Xn, 0 < n < m, are as required.

[a1] We have $(\mathfrak{M}_{a},\mathcal{A}^{e})\models\Gamma_{1}$ and so $\mathcal{A}_{2}(b)=\mathcal{A}_{2}^{e}(b)\in Z_{m}$ for all $b\in obj(\Gamma_{1})$. Hence, $\mathcal{A}_{2}(b)\in X_{m}=Z_{m}\cup W_{m}$ for all $b\in obj(\Gamma_{1})$.  
[a2] By the definition of disjoint unions and because { 20 a, A a } |= [ 1 ,
[a3] Firstly, we have, by the definition of c<nt and since c 20 is monotone (it distributes

oer intersections),
Zn +1 ⊆ Zn ∩ c 211 (Zn) ⊆ Xn ∩ c 211 (Xn).

(3)
Secondly, by the definition of disjoint unions, the first property of covering normal terms, and since c 277 is monotone

$$\bigcup_{n+1\leq i\leq m}W_{i}\subseteq\bigcup_{n\leq i\leq m}W_{i}\subseteq\bigcup_{n\leq i\leq m}W_{i}\cap e^{\mathfrak{M}}(\bigcup_{n\leq i\leq m}W_{i})\subseteq X_{n}\cap e^{\mathfrak{M}}X_{n}.$$
$$\mathbb{U}\left(X_{n}\right),$$
$$\left({4}\right)$$

From (3) and (4) we obtain

$$X_{n\,+1}\,=\,Z_{n\,+1}\,\cup\,\bigcup_{n\,+1\,\leq i\,\leq m}W_{i}\,\subseteq\,X_{n}\,\cap\,c^{\mathfrak{M}}\,X_{n}\,.$$
$$\left({\mathfrak{h}}\right)$$
$$\left[\mathrm{a}\,4\right]\ \mathrm{v}$$

[a4] We show that the three properties from Definition 46 are satisfied. Since

$$\left\{\,s u r_{1}\!\left(s\right)^{\mathfrak{M}_{m},\mathcal{A}_{m}}\,:\,s\,\in\,\Sigma_{1}\,\right\}$$

is a κ -partition of W m , we have | sur 1 ( s ) W m , A m | = κ for all s ∈ Σ 1 . This implies Property 1 since sur 1 (s) 201 , A ∩ W m = sur 1 (s) 201 , Am , W m ⊆ X m , and | X m | ≤ r .

Property 2 is an immediate consequence of the definition of E 1 . As for Property 3, we show that, for all w ∈ X m, we have w ∈ s W, A for an s ∈ Σ 1 . Fix a w ∈ X m. We distinguish two cases: firstly, assume w ∈ W m . Then, by the fact that { sur 1 ( s ) W m , Am s E E 1 } is a rc partition of Wm, it is clear that there exists an s E E 1 as required.

Secondly, assume w ∈ Z m = (c≤m ( sur 1 (√ Σ 1 ))) 2vα , Aα . By definition of c≤m t, we have w E (sur 1 (V Σ 1 )) 200 ,A α and so again ω E sur 1 (s) 201 ,A for some s E Σ 1 .

[a5] The proof is similar to that of Property [a4].