Theorem 3.3 K2 **is, under Lukasiewicz fuzzy logic, satisfiable by a witnessed**
model but not by a finite model.

Proof: **By the previous corollary it is enough to give a witnessed model of**
K2. One witnessed model of this fuzzy KB is the model I **defined by**
- ∆I = {1, 2, 3, . . . **} ∪ {∞}**,
- RIis the crisp relation {(i, i + 1) : i = 1, 2, 3, . . . **} ∪ {**(∞, ∞)}, - AI(∞) = 1 and AI(i**) = (2**i − 1)/2 ifor every i = 1, 2, 3**, . . .**
- a I **= 1.**
The fact that this model I satisfies K2 **can be easily checked by the reader, and**
the same for the first two conditions in the definition of witnessed model. On the other hand, the last condition in the definition of witnessed model is subtler, and we give here a proof. First of all we notice the following claim.

Claim 3.4 For every (complex) concept C and every ε > 0**, it holds one (and** only one) of the following two conditions:
Cond1(**C, ε**): C
I(∞) = 0, and there is some n ∈ N such that inf{C
I(i) : i ∈
N, i ≥ n} = 0 and sup{C
I(i) : i ∈ N, i ≥ n} ≤ ε.

Cond2(**C, ε**): C
I(∞) = 1, and there is some n ∈ N such that sup{C
I(i) : i ∈
N, i ≥ n} = 1 and inf{C
I(i) : i ∈ N, i ≥ n} ≥ 1 − ε.

Proof of Claim: **First of all, we notice that a trivial induction proves that**
C
I(∞) ∈ {0, 1} for every concept C**. Next we prove the claim by induction on**
the length of the concept C. The cases of ⊤ and an atomic concept A **are trivial.** Due to the interdefinability between connectives in the Lukasiewicz **case, it is** enough to study the connectives ¬, ⊓ and ∀R.C.

- The case of a concept ¬C **follows from the continuity of the Lukasiewicz**
negation; we remind the reader that continuity tells us that the function commutes with infima and suprema.

- Let us consider the case of a concept C ⊓ D. If both C and D **satisfy the**
second condition, then it is enough to see that Cond2(C, ε/**2) together with** Cond2(D, ε/2) implies Cond2(C ⊓ D, ε**); and this fact is a consequence of**
sup{(C ⊓ D)
I(i) : i ∈ N, i ≥ n} **= sup**{C
I(i) : i ∈ N, i ≥ n} ⊗ sup{DI(i) :
i ∈ N, i ≥ n} = 1 ⊗ 1 = 1 (by the continuity of ⊗**) and (1** −
ε 2
) ⊗ (1 −
ε 2
) ≥
1 − ε**. Let us consider now the case that at least one of the two concepts** satisfies the first condition. Without loss of generality, we can assume that Cond1(C, ε) holds. From this assumption it follows Cond1(C ⊓ **D, ε**)
because (C ⊓ D)
I(i) ≤ C
I(i**) for every** i.

- The case of a concept ∀R.C **follows from the fact that for every** i ∈ N,
(∀R.C)
I(i) = C
I(i **+ 1).**