The other system, as described in 19th-century textbooks, says this:

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[What if the digit in the subtrahend is bigger than the one in the minuend?]

Imagine adding 10 to both numbers. Obviously, the difference between them will not be changed.

But adding 10 to the digit in the subtrahend is the same as adding 1 to the digit immediately to its left.

So, add 10 to the [current] digit in the minuend, add 1 to the [next] digit in the subtrahend, and then perform the subtraction.