Alright, let's solve this problem step by step. We have 479903 and 18637, and we're subtracting the second number from the first.

Step 1: We'll start by subtracting the digit 7 and the borrow 0 from 3 in column 1 and get -4.
We add -4 to 10 and get 6 as the first digit of the result.

Step 2: We'll start by subtracting the digit 3 and the borrow 1 from 0 in column 2 and get -4.
We add -4 to 10 and get 6 as the next digit of the result.

Step 3: We'll start by subtracting the digit 6 and the borrow 1 from 9 in column 3 and get 2.
2 is the next digit of our result.

Step 4: We'll start by subtracting the digit 8 and the borrow 0 from 9 in column 4 and get 1.
1 is the next digit of our result.

Step 5: We'll start by subtracting the digit 1 and the borrow 0 from 7 in column 5 and get 6.
6 is the next digit of our result.

Step 6: We'll start by subtracting the digit 0 and the borrow 0 from 4 in column 6 and get 4.
4 is the next digit of our result.

So, 479903 - 18637 = 461266.