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Testing: Does the phantom period appear if I simplify it to this? $\forall\,a,b\in\pi_1(S)\cup \pi_2(S)\colon aSb\Leftrightarrow f(a) S' f(b)$. Yes. In this screen grab, see the period after $aSb$, but the period after $f(a) S' f(b)$ is missing.

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