[see also: just]
If we simply mimic the standard proof of ...... we are led to ......
If $M$ is empty we write simply $f\sim g$.
The rank of appearance (or simply the rank) of an integer in the sequence $U(P)$ is the least positive integer $n$ such that ......
The lower limit is defined analogously: simply interchange sup and inf in (1).
Then $G$ is simply $g$ with its periodic string read backwards.
The reason for preferring (1) to (2) is simply that (1) is manifestly invariant. [Note the double r in preferring.]
The presence here of the direct summand $H$ is simply to prevent $A$ from having disconnected spectrum.
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