Okay, let's tackle this math problem. We're starting with 599775 and subtracting 59117.

Step 1: We'll start by subtracting the digit 7 and the borrow 0 from 5 in column 1 and get -2.
We add -2 to 10 and get 8 as the first digit of the result.

Step 2: We'll start by subtracting the digit 1 and the borrow 1 from 7 in column 2 and get 5.
5 is the next digit of our result.

Step 3: We'll start by subtracting the digit 1 and the borrow 0 from 7 in column 3 and get 6.
6 is the next digit of our result.

Step 4: We'll start by subtracting the digit 9 and the borrow 0 from 9 in column 4 and get 0.
0 is the next digit of our result.

Step 5: We'll start by subtracting the digit 5 and the borrow 0 from 9 in column 5 and get 4.
4 is the next digit of our result.

Step 6: We'll start by subtracting the digit 0 and the borrow 0 from 5 in column 6 and get 5.
5 is the next digit of our result.

So, 599775 - 59117 = 540658.