Hello, the addition algorithm that is commonly used in the base 10 positional notation seems to make sense:

-we add the 1s with the 1s

-we add the 10s with the 10s

...

-when we reach at least 10 units in a position, we increment the following position and leave the remainder in the current position

yet can someone provide me with a proof that the addition algorithm guarantees that adding, for instance, 12 with 10 will yield the same result as 12 +1+1+1+1+1+1+1+1+1... (10 times)?

I know that 10 is the same as 1+1+1+1+1+1 (10 times) plus 12 = 1+1+1+1+1.... since the resulting number of 1s is the same. But what guarantees that this happens in the addition algorithm? What does it make to support this?

Thanks in advance