The grammatical genious of the German mind

After I checked out at the supermarket in Erfurt today, the check-out lady said to me, very quickly, "Brauchen Sie die Quittung?" Meaning to say, Do you need the receipt?

Here's what must have gone through her head, in just a few nano-seconds. This must have been her exact thought process:

"Quittung" ends in -ung, which means it is a feminine gender noun. So the feminine definite article would be "die." But here the noun "Quittung" is the direct object of the verb "brauchen," which would put it into the accusative case. However, the fact is that the definite article for singular feminine nouns in the accusative case is ALSO "die." So that's what I must apply.

And then she instantly put it all together and articulated the utterance. 

This is astoundingly quick logical thinking. This woman, in my opinion, should be teaching linguistic theory in some university. But here she is working as a check-out clerk in a supermarket. I just don't get it.