For those who haven’t been following the discussion in the comment thread, here’s the solution to that apparent hole in Picard’s Theorem…
Let us examine the argument slowly:
1. “ cannot take the value zero.” This is true. For suppose it did at some point . Then has the property that , which we know cannot happen.
2. “Since cannot take the value zero, cannot take the value .” This is pure nonsense. It appeals to one-to-oneness that does not hold, spectacularly. Since by Picard’s theorem, only does not take the value zero, it perfectly well can take the value (or any other non-zero even multiple of , which isn’t the year when Mozart was born), and .