Theorem 3.6. A valuation w *being consistent, i. e. satisfying the equality* w 0 = w*, entails the definite consistency of the associated decision of margin* η
—that is the partial truth assignment µη(w)— for any η *in the interval* 0 ≤ η ≤ 1.

Proof. Recall that the consistency of w is equivalent to its satisfying the inequalities (8). In accordance with the definition of definite consistency, we must show that for each C ∈ D and every p ∈ C , if all α ∈ C \ {p} are rejected for the decision of margin η associated with w, then p is accepted for this decision. Assume the contrary: p is not accepted, that is

$$w_{\overline{{{p}}}}\;\geq\;w_{p}-\eta.$$
$$(10)$$
wp ≥ wp − η. (9)
From (8) it follows that

$$w_{p}\,\geq\,\operatorname*{min}_{\begin{array}{c}{{\alpha\in C}}\\ {{\alpha\neq p}}\end{array}}w_{\overline{{{\alpha}}}}\,=\,w_{\overline{{{q}}}},$$
wα = wq, (10)
for some q ∈ C \ {p}. Let us fix such a q . By combining (9) and (10) we get wp ≥ wq − η. (11)
Now, since q 6= p, the hypothesis that q is rejected is saying that

$\left(12\right)^{2}$
$$\left(13\right)$$
$\left(14\right)^3$
wq − *η > w*q. (12)
On the other hand, (8) entails that

$$w_{q}\;\geq\;\operatorname*{min}_{\begin{array}{c}{{\alpha\in C}}\\ {{\alpha\neq q}}\end{array}}w_{\overline{{{\alpha}}}}\;=\;\operatorname*{min}{\Big(}\;w_{\overline{{{p}}}}\;,\;\operatorname*{min}_{\begin{array}{c}{{\alpha\in C}}\\ {{\alpha\not\in\{p,q\}}}\end{array}}w_{\overline{{{\alpha}}}}{\Big)},$$
$$w_{q}\;\geq\;w_{\overline{{{q}}}}-\eta,$$
, (13)
from which (11) and (10) —and the hypothesis that η ≥ 0— allow to conclude that wq ≥ wq − η, (14)
in contradiction with (12).

Corollary 3.7. For any η in the interval 0 ≤ η ≤ 1, the decision of margin η *associated with the upper revised valuation* v
∗ —that is the partial truth assignment µη(v
∗)*— is always definitely consistent.*
Proof. It is simply a matter of recalling that v
∗satisfies the hypothesis of the preceding theorem.