Why can we multiply by breaking up the factors as sums in different ways?

My friend and I were discussing some mathematical philosophy and how the number systems were created when we reached a question. Why can we multiply two different numbers like this?

Say we had to multiply $13\times 34$. One may break this up like $(10+3)\times (30+4)$. Applying distributive property here will give us the answer of 442. We can also choose to multiply this as $(6+7)\times (22+12)$. Intuitively, we can hypothesize that this should give us the same answer as $13\times 34$. How can we prove that our answer will be equal regardless of how we break up the numbers?

There are two ways I see to answer this question. One is from an axiomatic standpoint, where numbers are merely symbols on paper that are required to follow certain rules. The other uses the interpretation of multiplication as computing area. The former would take a good 5-10 pages to build up from the Peano axioms.

For the latter, you can draw a rectangle 13 units by 34 units. Break one side into 10 units and 3 units, and the other side into 30 units and 4 units. At this point, you should see that this decomposes the rectangle into four pieces, corresponding to the four terms you get from the distributive rule. The various was to compute $13 \cdot 34$ are all just ways to decompose the rectangle, and at the end of the day they all compute the same number: the area of the rectangle.

Well, that is literally what the distributive law tells you. It tells you that $$(a+b)(c+d)=a(c+d)+b(c+d)=ac+ad+bc+bd$$ and so whenever you break up the two factors of a product as a sum, you can use the "pieces" to compute the product.

So what you are really asking for is a proof of the distributive law itself. What constitutes a "proof" of such a basic fact depends heavily on what definitions of "numbers" and the operations on them that you are using (for some definitions, the distributive law is simply an axiom that you assume). But here is an intuitive explanation that works for natural numbers (and this can be turned into a rigorous proof if you define arithmetic of natural numbers in terms of cardinalities of sets).

We want to prove that $(a+b)c=ac+bc$. What does a product $xy$ of natural numbers mean? Well, it means you draw a grid of dots with $x$ rows and $y$ columns, and count up the total number of dots. So, to compute $(a+b)c$, you draw a grid with $a+b$ rows and $c$ columns. Now we observe that we can split such a grid into two pieces: the top $a$ rows and the bottom $b$ rows. The top $a$ rows form a grid with $a$ rows and $c$ columns, so they have $ac$ dots. The bottom $b$ rows form a grid with $b$ rows and $c$ columns, so they have $bc$ dots. In total, then, we have $ac+bc$ dots, so $(a+b)c=ac+bc$.

(In the computation of $(a+b)(c+d)$ above I used the distributive law in two different versions, one with the sum on the left side of the product and one with the sum on the right side of the product. You can prove the version with the sum on the right side of the product in the same way (you just split up the columns instead of the rows), or you can deduce it from the other version using commutativity of multiplication.)