[of sth, not: “on sth”]
......, the constant $C$ being independent of $n$.
Note that $F$ is independent of the choice of the family $S$.
Indeed, it is routine to verify that the index so constructed is independent of the choices made.
It turns out that this is independent of the representations taken (as long as they are faithful).
The two notions of rank are independent of each other.
a result of independent interest
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