Equivalence of Definitions of Connected Topological Space/No Separation iff No Clopen Sets

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Definition by No Clopen Sets implies Definition by Separation
Let $T$ be connected by having no clopen sets.

$T$ admits a separation, $A \mid B$ say.

Then both $A$ and $B$ are clopen sets of $T$, neither of which is either $S$ or $\O$.

From this contradiction it follows that $T$ can admit no separation

Definition by Separation implies Definition by No Clopen Sets
Let $T$ be connected by admitting no separation.

Suppose $\exists H \subseteq S$ which is clopen.

Then $\relcomp S H$ is also clopen.

Hence $H \mid \relcomp S H$ is a separation of $T$.

From this contradiction it follows that $T$ can have no non-empty proper subsets which are clopen.