 t$, then clearly $g(c)\preceq t_{\leftarrow}$. Else, $t\preceq x$ for *each* $t\in C_{g}$ shows that $t_{\leftarrow}\preceq c$ because $t_{\leftarrow}$ is the smallest of all the upper bounds $c$ of $C_{g}$. Hence $t_{\leftarrow}\in C_{g}$.

Property (ST3) for $C_{g}$ follows from a small yet significant modification of the above arguments in which the immediate successors $g(t)$ of $t\in C_{g}$ formally replaces the supremum $t_{\leftarrow}$ of $C_{g}$. Thus given a $c\in C$, if there is *some* $t\in C_{g}$ for which $g(c)\preceq t$ then $g(c)\prec g(t)$; this combined with $(c=t)\Rightarrow(g(c)=g(t))$ yields $g(c)\preceq g(t)$. On the other hand, $t\prec c$ for *every* $t\in C_{g}$ requires $g(t)\preceq c$ as otherwise $(t\prec c)\Rightarrow(c\prec g(t))$ would, from the resulting consequence $t\prec c\prec g(t)$, contradict the assumed hypothesis that $g(t)$ is the immediate successor of $t$. Hence, $C_{g}$ is a $g$-tower in $X$.

To complete the proof that $g(c)\in C_{\textrm{T}}$, and thereby the argument that $C_{\textrm{T}}$ is a tower, we first note that as $_{\rightarrow}T$ is the smallest tower and $C_{g}$ is built from it, $C_{g}=\,_{\rightarrow}T$ must infact be $_{\rightarrow}T$ itself. From Eq. (\[Eqn: chain\_g\]) therefore, for every $t\in\,_{\rightarrow}T$ either $t\preceq g(c)$ or $g(c)\preceq t$, so that $g(c)\in C_{\textrm{T}}$ whenever $c\in C_{\textrm{T}}$. This concludes the proof that $C_{\textrm{T}}$ is actually the tower $_{\rightarrow}T$ in $X$.

From (ST2), the implication of the chain $C_{\textrm{T}}$ $$C_{\textrm{T}}=\,_{\rightarrow}T=C_{g}\label{Eqn: ChainedTower}$$ being the minimal tower $_{\rightarrow}T$ is that the supremum $t_{\leftarrow}$ of the totally ordered $_{\rightarrow}T$ *in its own tower* (as distinct from in the tower $X$: recall that $_{\rightarrow}T$ is a subset of $X$) must be contained in itself, that is $$\sup_{C_{\textrm{T}}}(C_{\textrm{T}})=t_{\leftarrow}\in\,_{\rightarrow}T\subseteq X.\label{Eqn: sup chain}$$ This however leads to the contradiction from (ST3