hence, the Zadeh, Lukasiewicz (note that in Lukasiewicz logic, C → D **coincides**
with ¬C ⊔ D**) and G¨odel families given in Table 1 satisfy the assumption in the**
second item of the following result.

Lemma 4.2 (Assuming ⊗ is continuous) Let K be a fuzzy KB and let **C, D** be two concepts.

1. The following statements are equivalent:
- K ∪ {hC ⊑ D ≥ 1i} **is satisfiable,**
- K ∪ {C ≡ A ⊓ D}, where A **is a new atomic concept, is satisfiable.**
This equivalence also holds when we consider satisfiability **in a finite model.**
2. (Assuming the constructor min **is definable) The following statements are**
equivalent:
- K ∪ {hC ⊑ D ≥ 1i} **is satisfiable,**
- K∪{C ≡ min{A, D}}, where A **is a new atomic concept, is satisfiable.**
This equivalence also holds when we consider satisfiability **in a finite model,** and the same for satisfiability in a witnessed model.

Proof: **Let us consider the first item. One direction is a consequence of**
the fact that t-norms satisfy x ⊗ y ≤ y**. For the other, let us point out that** continuous t-norms are divisible (see [11, Lemma 2.1.10]) in the sense that for every x, y ∈ [0, 1] there is an element a such that a⊗x = min{x, y}**; this element** a can be defined as sup{z ∈ [0, 1] : x ⊗ z ≤ y} **and is commonly denoted** x ⇒ y. Hence, we can extend any interpretation I **to encompass the new atomic concept**
A defining AI(x**) :=** DI(x) ⇒ C
I(x).

One direction of the second item follows from the fact that min{x, y} ≤ y.

The other is proved extending any interpretation I **to encompass the new atomic**
concept A defining AI(x**) :=** C
I(x**). Q.E.D.**

## Lemma 4.3 (Assuming ⊗ **Is Continuous)**

1. There is an algorithm that converts every unfoldable fuzzy KB K = hA, T i into a fuzzy ABox A⋆**in such a way that**
K is satisfiable iff A⋆**is satisfiable.**
The algorithm also preserves satisfiability in a finite model.

2. (Assuming the constructor min **is definable) There is an algorithm like**
in the previous item which additionally also preserves satisfiability in a witnessed model.