If \(f\) and \(g\) are two partial functions from a space \(S\) to \(R\), one writes \(f\subseteq g\) to mean that whenever \(f(x)\simeq w\) then also \(g(x)\simeq w\). Then for two partial relations \(\bP\) and \(\bQ\) of the same arity, \(\bP\subseteq\bQ\) means that whenever \(\bP(\bd)\) is defined, so is \(\bQ(\bd)\), and both have the same truth value. We can extend this to \(k\)-tuples by setting \(\langle\bP_1, \ldots, \bP_k\rangle\subseteq\langle\bQ_1, \ldots, \bQ_k\rangle\) if \(\bP_i\subseteq\bQ_i\) for each \(i\). It is not terribly difficult to show that the functional \([E]\) defined above, and based on \eqref{Eexample}, has the monotonicity property: if \(\langle\bP_1, \ldots, \bP_k\rangle\subseteq\langle\bQ_1, \ldots, \bQ_k\rangle\) then \([E](\langle\bP_1, \ldots, \bP_k\rangle)\subseteq[E](\langle\bQ_1, \ldots, \bQ_k\rangle)\). There is a very general theory of monotone mappings like this, from which it follows that \([E]\) does have a fixed point. Moreover, if there are more than one then there is a unique one that is least, in the sense that it is in the \(\subseteq\) relation to any other. This least fixed point is precisely the best candidate we mentioned above. It contains the information that any fixed point must have.