= Five values force infinitely many simple preimages
{title2=$\sum\overline N\le\tfrac12\sum N+\tfrac12\sum N_{\rm simple}$}
If a transcendental <meromorphic function> had only finitely many simple preimages over each of five distinct finite values, their simple counting functions would be $O(\log r)$. The displayed <multiplicity> inequality and the <Nevanlinna first main theorem> would bound the truncated sum by $5T_f/2+O(\log r)$. The <Nevanlinna second main theorem> requires at least $3T_f$ up to its $o(T_f)$ error, a contradiction. Thus at least one value has infinitely many simple preimages; four other values can remain totally ramified.
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