User:Dfeuer/Minimal Enclosing Complete Lattice is Order Completion

Theorem
Let $(T,\le)$ be a complete lattice.

Let $S \subseteq T$.

Suppose that for any $U$ such that $S \subseteq U \subsetneqq T$,


 * $(U, \le)$ is not complete.

Then $T$ is an order completion of $S$.

Proof
Let $(V,\preceq)$ be a complete lattice containing $S$ such that $\preceq\restriction S=\le\restriction S$.

Define $f_n$ recursively