The IP addresses will not match. The IP address identifies the email server and not the client.
Moderator Note: This answer was originally provided by Mshen. I’ve moved it here from the original post.
Actually there was more to the answer – the headers above may or may not show the client’s IP address. It depends on the receiving system and what addresses are shown in the headers. To read the headers for IP address traversals, start at the bottom and work up to the top.