Q: How can you check if all your photos in Aperture are included in a back-up?

If your Aperture library is on your system drive or on another drive that is included in your Time Machine backup, your Aperture library will be included in the Time Machine backup.

Time Machine is working well with Aperture 3.4.5.

Aperture vaults are a legacy item from the times, when Time Machine did not work well with photo libraries. You can use them to create a second backup of your photos on a different drive.  A vault is essentially a copy of your Aperture Library, and when you update a vault, all changes since the last backup will be applied to this copy.  It is an incremental backup and It is faster then copying your library entirely over and over again.

I back up to a WD My Book, is there a way of seeing these photos when I go into the back up after clicking on the MyBook?

To view the photos in the backup you restore the library from the backup using Time Machine.

I don't create vaults.  I make occasionally a copy of the Aperture library on a different drive in addition to my regular Time Machine backup.

This manual page explains how to create vaults, if you would like to use a vault:  http://documentation.apple.com/en/aperture/usermanual/index.html#chapter=27%26se ction=0

To restore your Aperture Library from Time Machine select the Aperture Library in the Funder and then enter Time Machine.

In the star wars display go back in time to the version you want and click "restore to" to restore to a different drive, if you want to verify the backup.

The Time Machine backup is incremental. You can open the Masters folder of the backup from the Finder and look at individual photos (if you are ver careful not to mess up the backup while browsing), but the Photos app cannot one the backup directly on the Time Machine drive.  And looking manually will not give you the certainty that the backup is working. You can see, if all your original photos are there, but not if the edits and metadata or your albums and projects can be restored correctly.  the backup is a complicated package of linked items.

If you select your Time Machine backup disk in the Finder, you will see a folder backup.backupdb at the top level.  If you click it and wait a bit, you can now see folders for your computers and the different backup dates. Select one of them and open it. You will see the folders for  your Mac mirrored and can select the Photos Library in the backed up Pictures folder.  Ctrl-click the Photos Library there and select "Show Package Contents". You will see nested folders sorted by year-month-day with the original images you imported. Just look to convince yourself your photos are there.  This is no way way to restore photos.