 $\textrm{Multi}(X)$ and $$\mathcal{X}=\{ C\in\mathcal{P}(F)\!:C\textrm{ is a chain in }(F,\preceq)\}\subseteq\mathcal{P}(F)$$ denote the corresponding Hausdorff level of the chains of $F$, with $$\mathcal{C}_{\textrm{T}}=\{ C_{\alpha},C_{\beta},\cdots,C_{\nu},\cdots\}=\,_{\rightarrow}\mathcal{T}\subseteq\mathcal{P}(F)$$ being a $g$-sequential chain in $\mathcal{X}$ . **By Hausdorff Maximal Principle, there is a maximal fixed-point $g$-towered chain $C_{\leftarrow}\in\mathcal{X}$ of $F$ $$\sup_{\mathcal{C}_{\textrm{T}}}(\mathcal{C}_{\textrm{T}})=C_{\leftarrow}=\{ f_{\alpha},f_{\beta},f_{\gamma},\cdots\}=g(C_{\leftarrow})\in\mathcal{C}_{\textrm{T}}\subseteq\mathcal{P}(F).$$

Zorn’s Lemma now applied to this maximal chain yields its supremum as the maximal element of $C_{\leftarrow}$, and thereby of $F$. It needs to be appreciated, as in the case of the algebraic Hamel basis, that the existence of this maximal non-functional element was obtained purely set theoretically as the “limit” of a net of functions with increasing non-linearity, without resorting to any topological arguments. Because it is not a function, this supremum does not belong to the functional $g$-towered chain having it as a fixed point, and this maximal chain does not possess a largest, or even a maximal, element, although it does have a supremum.[^24] The supremum is a contribution of the inverse functional relations $(f_{\alpha}^{-})$ in the following sense. From Eq. (\[Eqn: func-multi\]), the net of increasingly non-injective functions of Eq. (\[Eqn: chaos1\]) implies a corresponding net of increasingly multivalued functions ordered inversely by the inverse relation $f_{\alpha}\preceq f_{\beta}\Leftrightarrow f_{\beta}^{-}\preceq f_{\alpha}^{-}$. Thus the inverse relations which are as much an integral part of graphical convergence as are the direct relations, have a smallest element belonging to the multifunctional class. Clearly, this smallest element as the required supremum of the increasingly non-injective tower of functions defined