40 **M. Slota and J. Leite**
By the inductive assumption we obtain that T
n−1 PM⊕T A
(I) = T
n−1 PM⊕T A′ (I**), so**
T
n PM⊕T A(I) = mod n H∗(r)
r ∈ PM ∧ T
n−1 PM⊕T A′ (I) |= B(r)
o⊕
T A
.

Corollary D.2 **now implies that**
T
n PM⊕T A(I) = mod n H∗(r)
r ∈ PM ∧ T
n−1 PM⊕T A′ (I) |= B(r)
o⊕
T A
′
= T
n PM⊕T A′ (I) .

Proposition 4.7. Let T be a TBox, A an ABox and M **an MKNF interpretation.**
Then M is a minimal change dynamic stable model of ∅ ⊕T A **if and only if** M =
mod(**T ∪ A**).

Proof of Proposition 4.7 By Proposition 3.2 and Theorem A.12, M **is a minimal change dynamic stable**
model of ∅ ⊕T A **if and only if**

$$M=\bigcap_{n\geq0}T_{{\mathcal{P}}^{M}\oplus{\mathcal{T}}{\mathcal{A}}}^{n}({\mathcal{I}})$$

where T
0 PM⊕T A(I) = I
T
1 PM⊕T A(I) = mod(H∗(r) r ∈ PM **∧ I |**= B(r)	⊕
T A) = mod(∅ ⊕T A)
T
n PM⊕T A(I) = T
1 PM⊕T A(I) for all n > 1 So M is a minimal change dynamic stable model of ∅ ⊕T A **if and only if** M =
mod(∅ ⊕T A**). Further,**
mod(∅ ⊕T A) = incorporateT(A, mod(∅**)) =** incorporateT(A, I) = mod(**T ∪ A**) .

Hence, M is a minimal change dynamic stable model of ∅ ⊕T A **if and only if**
M = mod(T ∪ A).