A meromorphic essential singularity omits at most two sphere values

Suppose three distinct sphere values each failed to occur infinitely often in some punctured neighborhood of $a$. After passing to a common smaller neighborhood and then shrinking past their finitely many preimages, all three values are omitted. By [L2], choose a Möbius transformation $M$ sending them to $0$, $1$, and $\mathrm{\infty }$. Then $g:=M\circ f$ is holomorphic on that smaller punctured disc and still has an essential singularity at $a$, because a biholomorphic target change cannot turn an essential singularity into a removable singularity or pole.

The function $g$ omits the finite values $0$ and $1$, so [L1] under [A1] gives a contradiction. Thus at most two sphere values can fail the infinitely-often property, and every other sphere value occurs infinitely often in every punctured neighborhood.

This is the meromorphic Great Picard conclusion.