The local set of labeled elemental constraints {lec1, lec2, ... , leck}is overall consistent iff the setunion of their associated label sets (∪i=1,k{labeli}) is not an I-L-Set.

Proof: The label set (∪i=1,k{labeli}) is the support-set of the simultaneous fulfillment of {lec1, lec2,
--- , leck}. Moreover, the set of I-L-Sets is complete and sound with respect to overall TCN (Theorem 8), such that any label set not in the set of I-L-Set is overall consistent. Therefore (*Theorem 2)*,
(∪i=1,k{labeli}) and {lec1, lec2, ... , leck} are overall consistent iff ∪i=1,k{labeli} is not an I-L-Set. ◊
Definition 5 *(Labeled-consistency* 3): Let's assume a labeled-TCN of n nodes (and the corresponding complete set of I-L-Sets) and a set of k (1≤k≤( 2 n)) constraints, each one of which is between any pair of nodes in the TCN:
{cij} / 1≤i≤n, 1≤j≤n, i≠j.

The set of constraints {cij} is *labeled-consistent* with respect to the nodes involved in these constraints, iff:

i) For each constraint cij, there exists an elemental labeled constraint elcij.x between (ni, nj) in the TCN such that elcij.x satisfies cij. That is: ∀cij, ∃elcij.x∈lcij / cij ⊕ ecij.x ≠ ∅.

ii) The resulting set of the union of label sets associated to these elemental labeled constraints
(which satisfy {cij}) is not an I-L-Set: U ij ij.x c ∀ {label } is not an I-L-Set. Note that this is the condition of Theorem 9. ◊
Theorem 10. Let's assume a labeled-TCN of n nodes (and the corresponding complete set of I-LSets) and a set of k (1≤k≤( 2 n)) constraints, each one of which is between any pair of nodes in the TCN:
{cij} / 1≤i≤n, 1≤j≤n, i≠j.

The set of constraints {cij} is overall consistent iff {cij} is labeled-consistent with respect to the nodes involved in constraints {cij}.

Proof: The proof is trivial according to *Definition 5* and *Theorem 9*. We have that the set of constraints {cij} is consistent iff there exists a local set of elemental constraints in the TCN {elcij.x}
that makes {cij} labeled-consistent (*Definition 5*). Thus, the local set {elcij.x} is consistent (Theorem 9), such that {cij} is also consistent. ◊
For instance, we can determine whether any pair of constraints c'ij and c'kl can hold simultaneously
(that is, they are overall consistent) if:
∃elcij.x∈lcij / c'ij ⊕ ecij.x≠∅ ∧ ∃elckl.y∈ckl / c'kl ⊕ eckl.y≠∅ ∧ {labelij.x}∪{labelkl.y}
is not an I-L-Set.

Moreover, any local instantiation of any k-1 (1<k≤n) variables {t1=v1, t2=v2, ..., t(k-1)=v(k-1)} can be extended to a global solution if:
∃elc10.x∈lc10 / v1∈ec10.x,...... , ∃elc(k-1)0.y∈lc(k-1)0 / v(k-1)∈ec10.x, where lci0 is the constraint between ni and T0, and {label10.x}∪{label20.y}∪ .... ∪{label(k-1)0.y}is not and I-L-Set.