A) (a) = A2(a), for a E obj(F), and A) (at) E t 20,A, for all t E Σ 1 (we can choose an injective function for A 1 since the sets t 20,A are infinite).

As in the corresponding part of the proof of Theorem 20, it can show by induction that sur1(t)2111,A1 = t211,A for all t E term(F).

Let us see now why ( 201 , A 1 ) |= F 1 follows from this equation. For R( a, b ) E F 1 we have R(a,b) ∈ Γ and so ( 20,A) |= R(a,b) . Hence ( 201,A 1 ) |= R(a,b) . We have ( 20,A) |=
(V Σ 1 ) = T (by the definition of Σ 1 ). Hence ( W 1 , Α 1 ) |= sur 1 (V Σ 1 ) = T and so, by the definition of c<m, <M 1 , A 1 ) |= (c<m(sur 1 (V Σ 1 ))) = T.  It remains to observe that A 2 (a) E sur 1 (t a) 2011 A 1 for all a'E obj (Γ), A 2 (a) E sur 1 (s) 2011 A 1 whenever (a : s) E Γ 1 and A 2 (a t ) E sur 1 (t) 2011 A 1 for all t E I 1 .

The construction of a model in M 2 satisfying Γ 2 is similar and left to the reader.

It remains to show the implications (2) ⇒ (1) and (3) ⇒ (1). They are similar, so we concentrate on the first. In the proof of Theorem 20 it was possible to construct the required model for Γ by merging models for Γ 1 and Γ 2 . The situation is different here. It is not possible to merge models for Γ 1 and Γ 2 in one step, since we do not know whether they satisfy sur 1 (V Σ 1 ) = T and sur 2 (V Σ 1 ) = T 1 respectively. We only know that they satisfy the approximations a : sur 1 (s) Ac ≤ m (sur 1 (V ∑ 1 )) and a : sur 2 (s) Ad ≤ r (sur 2 (V ∑ 1 )),
respectively, for a : s E F. To merge models of this type we have to distinguish various pieces of the models and have to add new pieces as well. To define those pieces we need a technical claim. As in the proof of Theorem 17, take cardinals κ i , i ∈ {1, 2} as in Lemma 45 for ( L i , M i ) and put κ = max { κ 1 , κ 2 } .

Claim 1. Suppose (2) holds.

(a) There exist M 1 = ( W 1 , F M , R M ) E M 1 , an assignment A = ( A 1 , A 2 ) into M 1 , and a sequence Xo,..., Xm of subsets of W1 such that
[a1] A2(a) E Xm, for all a E obj(F1),
[a 2]   ( M 1 , A ) 
[a3] Xn+1 ⊆ Xn ∩ c 201 (Xn), for all 0 ≤ n < m,
[a4] The set {sur1(s) 2011 . A N X m : s E E1} is a k -partition of X m :
[a 5] The sets
{sur1(s)201:A n(Xn - Xn+1): s E Σ 1 }
are ~-partitions of Xn - Xn+1, for 0 ~ n < m.

[a 6] | W 1 - X 0 | = κ.

(b) There exist 𝑠𝑠 2 = 〈 W 2 , G 2 V , Q 2 V ⟩ ∈ M 2 , an assignment B = 〈 B 1 , B 2 〉 , and a sequence Yo,..., Y, of subsets of W2 such that
[b1] B2(a) E Y, , for all a E obj(F 1 ),
[b2]  (M2, B)  =