Using your Head is Permitted

This month, I take this trend to its extremes, asking you to prove something even more obvious:

Prove that if two box-shaped parcels are placed one inside the other, the product length*width*height of the outside parcel is at least as large as that of the inside parcel.

This is, indeed, a very obvious fact. Here's one proof for it:

The length*width*height of a box-shaped parcel is its volume. The difference between the volume of the outside parcel and the volume of the inside parcel is the volume of all air trapped between the two parcels. As this is the volume of an actual geometrical shape, it cannot be negative. Q.E.D.

This month's challenge is to prove that the volume of the outside parcel is at least as large as the volume of the inside parcel by a proof that is different than the argument presented here. (This is a challenge unlike normal Using your Head is Permitted riddles, where any working proof method is accepted.) I intend to be very broad in my interpretation of what consists of a proof using the same argument as the one presented above. It is an argument that can be presented in many ways, and concealed quite cunningly. The challenge is to find a completely unrelated proof.

Hint: If you're using topology, set theory or calculus, you are in murky waters already. Look for a proof that (like last month's solution) relies solely on linear algebra.

As in last month's riddle, your proof should be applicable to parcels with any number of Euclidean dimensions.

List of solvers: