Okay, we are tasked with adding 9513 and 586. Let's begin!

Step 1: We'll start by adding the digits 3 & 6 in column 1 and get 9.

Step 2: We'll start by adding the digits 1 & 8 in column 2 and get 9.

Step 3: We'll start by adding the digits 5 & 5 in column 3 and get 10.
We'll write down the last digit 0 and carry the 1 to the next column.

Step 4: We'll start by adding the digits 9 & 0 in column 4 and get 10.
We'll write down the last digit 0 and carry the 1 to the next column.

So, 9513 + 586 = 10099.

