Claim 3.8 For all (complex) concepts C**, one (and only one) of the following**
conditions holds:
- 0 = C
I(i) **for every** i ∈ N,
- 0 < CI(1) ≤ C
I(2) ≤ C
I(3) ≤ . . . and sup{C
I(i) : i ∈ N} = 1.

Proof of Claim: **This can be straightforwardly proved by induction on the**
length of the concept C **using how the fuzzy operators involved in the product**
case are defined5.

- The case of a concept ¬C **is a trivial consequence of the fact that Product**
negation is the one given in Table 1 (usually called strict negation).

- The cases of concepts built with ⊓ and ⊔ **follow from the monotonicity of**
the functions ⊗ and ⊕.
- The cases of concepts built with ∀R. and ∃R. **follow from the fact that for**
every i ∈ N, (∀R.C)
I(i) = C
I(i + 1) = (∃R.C)
I(i).

This finishes the proof of the claim. q.e.d.

Now it is time to use this previous claim in order to check that for every **pair**
of concepts C and D there is some i ∈ N **such that (**C ⊑ D)
I = C
I(i) ⇒ DI(i).

By the previous property we can distinguish three different cases, **which cover** all possibilities.

- **Firstly, we consider the case where 0 =** C
I(i) for every i ∈ N**. Then, it is**
trivial that (C ⊑ D)
I = inf{0 ⇒ DI(i) : i ∈ N} **= 1 =** C
I(i) ⇒ DI(i**) for**
every i ∈ N.

- **Secondly, we consider the case where 0** 6= C
I(i) and 0 = DI(i**) for every**
i ∈ N**. Then, it is trivial that (**C ⊑ D)
I **= inf**{C
I(i) ⇒ 0 : i ∈ N} **= 0 =**
C
I(i) ⇒ DI(i**) for every** i ∈ N.
- Lastly, we consider the case where 0 < CI(1) ≤ C
I(2) ≤ C
I(3) ≤ **. . .**,
sup{C
I(i) : i ∈ N} = 1, 0 < DI(1) ≤ DI(2) ≤ DI(3) ≤ . . . and sup{DI(i) : i ∈ N} **= 1. In order to finish this proof it suffices to consider the case that (**C ⊑ D)
I 6**= 1, because if (**C ⊑ D)
I **= 1 then it**
is clear that (C ⊑ D)
I = C
I(i) ⇒ DI(i) for every i ∈ I**. Hence, let**
us assume that (C ⊑ D)
I 6= 1. Thus, there is some i0 ∈ N **such that**
(C
I(i0) ⇒ DI(i0**)) =** DI (i0)
CI (i0) < **1. On the other hand, by continuity we**
know that

$$\lim_{i\to\infty}\min\left\{\frac{C^{\mathcal{I}}(i)}{D^{\mathcal{I}}(i)},\frac{D^{\mathcal{I}}(i)}{C^{\mathcal{I}}(i)}\right\}\;=\;\min\left\{\frac{\lim_{i\to\infty}C^{\mathcal{I}}(i)}{\lim_{i\to\infty}D^{\mathcal{I}}(i)},\frac{\lim_{i\to\infty}D^{\mathcal{I}}(i)}{\lim_{i\to\infty}C^{\mathcal{I}}(i)}\right\}\;=\;1$$

5Here, as opposed to the Lukasiewicz case, the fact that the implication operator ⇒ **is not**
involved in the construction of complex ALC **concepts is crucial.**