To prove (i), note that since S
′is disjoint from S, and Y is a subset of X ∩ S,
Z ∩ S = X ∩ (S
′ ∪ Y ) ∩ S = X ∩ Y ∩ S = (X ∩ S) ∩ Y = Y.

To prove (ii), note that set Z is clearly a subset of X. It cannot be equal to X,
because otherwise we would have, by (i),

$$-\mathbf{\nabla}\times\mathbf{B}$$
$=\;\frac{1}{1+\sqrt{2}}$ 3. 
Y = Z ∩ S = X ∩ S;
this is impossible, because Y is a proper subset of X ∩ S. Property (iii) follows from Lemma 4, because X |= Γ2, and S
′ ∪ Y contains all head atoms of Γ2.

Since X is a stable model of Γ1 ∪ Γ2, from property (ii) we can conclude that Z 6|= (Γ1 ∪ Γ2)
X . Consequently, by property (iii), Z 6|= ΓX
1
. Since all atoms that occur in Γ1 belong to S, Γ
X
1 = ΓX∩S
1, so that Z 6|= ΓX∩S
1. Since all atoms that occur in Γ
X∩S
1belong to S, it follows that Z ∩ S 6|= ΓX∩S
1. By property (i), we conclude that Y 6|= ΓX∩S
1.

Proposition 10 (Splitting Set Theorem). Let Γ1 and Γ2 be two theories such that no atom occurring in Γ1 is a head atom of Γ2. Let S *be a set of atoms containing* all head atoms of Γ1 but no head atoms of Γ2. A set X *of atoms is a stable model of* Γ1 ∪Γ2 iff X ∩S is a stable model of Γ1 and X *is a stable model of* (X ∩S)∪Γ2.

Proof. We first prove the claim in the case when S contains all atoms of Γ1. If X ∩ S is not a stable model of Γ1 then X is not a stable model of Γ1 ∪ Γ2 by Lemma 7. Now suppose that X ∩ S is a stable model of Γ1. Then, by Lemma 6, Γ
X∩S
1is equivalent to X ∩ S. Consequently,

$$\begin{array}{r c l}{{(\Gamma_{1}\cup\Gamma_{2})^{X}\;=\;\Gamma_{1}^{X}\cup\Gamma_{2}^{X}}}&{{=\;\Gamma_{1}^{X\cap S}\cup\Gamma_{2}^{X}\;\leftrightarrow\;(X\cap S)\;\cup\;\Gamma_{2}^{X}}}\\ {{}}&{{=\;(X\cap S)^{X}\cup\Gamma_{2}^{X}\;=\;\left((X\cap S)\cup\Gamma_{2}\right)^{X}}}\end{array}$$

We can conclude that X is a stable model of Γ1 ∪ Γ2 iff X is a stable model of Γ2 ∪ (X ∩ S).

The most general case remains. Let S1 be the set of all atoms in Γ1 (the value of S for which we have already proved the claim). In view of the special case described above, it is sufficient to show that, for any set S of atoms that respects the hypothesis conditions, X ∩ S1 is a stable model of Γ1 and X is a stable model of (X ∩ S1) ∪ Γ2 (34)
holds iff X ∩ S is a stable model of Γ1 and X is a stable model of (X ∩ S) ∪ Γ2. (35)