before

The sum of the depths is at most two-thirds of what it was before.

Each tree is about two-thirds as deep as it was before.

The remainder of the estimation is largely as before with $B$ replaced by $C$.

The proof concludes as before.

Most of the theorems presented here have never been published before.

As noted before, there exists $N$ homeomorphic to $P$ such that ......

Note that $P$ comes before $Q$ along the arc $l$.

We may require that the point $P$ lie in one of the trees constructed before or during the $i$th stage of the induction. [Note the subjunctive lie.]



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