36 **M. Slota and J. Leite**
TBox T and ABox A and M **an MKNF interpretation such that for every subset**
S of the set { H∗(r) | r ∈ P ∧ M |= B(r) } **the following condition is satisfied:**
mod(S ⊕
T A) = mod(S ∪ O) .

Then M is an MKNF model of hO,Pi if and only if M **is a minimal change dynamic**
stable model of P ⊕T A.

Proof of Proposition 4.3 By Propositions C.10 and C.9, M is an MKNF model of hO,Pi **if and only if**

$$M=\bigcap_{n\geq0}T_{\langle{\mathcal{O}},{\mathcal{P}}^{M}\rangle}^{n}({\mathcal{I}})$$
(I**) (D3)**
where for any set of first-order interpretations N **we have**
ThO,PMi(N) = mod O ∪ H
∗(r) r ∈ PM ∧ N |= B(r)	 .

On the other hand, by Proposition 3.2 and Theorem A.12, M **is a minimal change**
dynamic stable model of P ⊕T A **if and only if**

$$M=\bigcap_{n\geq0}T_{{\mathcal{P}}^{M}\oplus{\mathcal{T}}{\mathcal{A}}}^{n}({\mathcal{I}})$$
PM⊕T A(I**) (D4)**
where for any set of first-order interpretations N **we have**
TPM⊕T A(N) = mod  H∗(r) r ∈ PM ∧ N |= B(r)	⊕
T A.

Suppose now that M is an MKNF model of hO,Pi**. Then from (D3) and Lemma**
B.4 we obtain that for every n ∈ N **that**
nH
∗(r)
r ∈ PM ∧ T
n hO,PMi
(I) |= B(r)
o⊆ { H
∗(r) | r **∈ P ∧** M |= B(r) } .

Hence, by the assumption of the proposition,

$$\mathsf{mod}\left(\left\{\,H^{*}(r)\,\left|\,\,r\in\mathcal{P}^{M}\wedge T^{n}_{(\mathcal{O},\mathcal{P}^{M})}(\mathcal{I})\models B(r)\,\right.\right\}\oplus^{\mathcal{T}}\mathcal{A}\right)$$ $$=\mathsf{mod}\left(\mathcal{O}\cup\left\{\,H^{*}(r)\,\left|\,\,r\in\mathcal{P}^{M}\wedge T^{n}_{(\mathcal{O},\mathcal{P}^{M})}(\mathcal{I})\models B(r)\,\right.\right\}\right)$$ (D5)
* [10] A. A. K.  
By induction on n **we will now prove that** T
n
hO,PMi
(I) = T
$(\mathcal{T})=T^n{}_{M^1,n}=\left(\mathcal{T}\right)$. 
PM⊕T A
(I).

1
◦ For n **= 0 we have**

$$T_{\langle{\mathcal{O}},{\mathcal{P}}^{M}\rangle}^{0}({\mathcal{I}})={\mathcal{I}}=T_{{\mathcal{P}}^{M}\oplus{\mathcal{T}},{\mathcal{A}}}^{0}({\mathcal{I}})$$
 $2^{\circ}$ We assume the claim holds for $n-1$, i.e. 
$$a=1,{\mathrm{i.e.}}$$
$$T_{\langle{\mathcal{O}},{\mathcal{P}}^{M}\rangle}^{n-1}({\mathcal{I}})=T_{{\mathcal{P}}^{M}\oplus}^{n-1}\tau_{{\mathcal{A}}}({\mathcal{I}})$$
$$(\mathrm{D}6)$$
(I) (D6)