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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