

![0_image_1.png](0_image_1.png)

![0_image_0.png](0_image_0.png)

$$\left(\,8\right)$$

```
  We nd, now using [b5℄, [b6℄, and Lemma 47 (1), models Vi =
                                          
                                           D
                                            
                                            Bi; FVi; RVi
                                                    E
                                                     
                                                     2 M1
                                                         
with assignments Ai =
              
               Ai
                1; Ai
                   2
                   
                    , 1  i  r  1, su
                                 h that, for 0  i  r  1,
                                                 

```

$$s u r_{1}\!\left(s\right)^{\mathfrak{M}_{i},A_{i}}=s u r_{2}\!\left(s\right)^{\mathfrak{M}_{2},B}\cap B_{i}\,\,\mathrm{for~all}\,\,s\in\Sigma_{1}.$$
Let  $$\mathfrak{M}_{1}^{j}=\left\langle W_{1}\cup(W_{2}-Y_{r}),\mathcal{F}^{\mathfrak{M}_{1}},\mathcal{R}^{\mathfrak{M}_{1}^{j}}\right\rangle\in\mathcal{M}_{1}$$  be the disjoint union of the $\mathfrak{M}^{i}$, $-1\leq i<r$, and $\mathfrak{M}_{1}$, and let 

$$\mathfrak{M}_{2}^{l}\,=\,\left\langle\,W_{2}\,\cup\,\left(\,W_{1}\,-\,X_{m}\,\right),\,\mathcal{G}^{\mathfrak{M}_{2}},\,\mathcal{Q}^{\mathfrak{M}_{2}^{l}}\,\right\rangle\,\in\mathcal{M}_{2}$$

be the disjoint union of the Wi
, 1  i < m, and W2. We assume Xm = Yr and so the domain of both ADMs is W1 [ W2.

```
Dene a model W =
               
                 W; (F [ G)W; (R [ Q)W
                                     2 M based on W = W1 [W2 by putting

```

$$\mathbf{\partial}\,\,\mathcal{R}^{\,\mathfrak{M}}=\mathcal{R}^{\,\mathfrak{M}_{1}},$$

$$\bullet\ {\mathcal{F}}^{\mathfrak{M}}={\mathcal{F}}^{\mathfrak{M}_{1}},$$

$${\textbf{\textit{Q}}}^{\mathfrak{M}}=\,Q^{\mathfrak{M}_{2}},$$

$$\bullet\ \mathcal{G}^{\mathfrak{M}}=\mathcal{G}^{\mathfrak{M}_{2}}\,.$$
