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Here’s the procedure that I followed on both an iPhone 4 and 4s when I had to deal with the iPhone Upgrade Verification Failure.
iPhone Update verification failure
I found this solution by accident as you will see.

Note: If you have a good charge, you don’t have to connect to a power source.

  1. Connect to a wireless router – This will enable the Download button
  2. Settings > General > Software Update
  3. The download will take about 10-15 minutes
  4. When the download is finished you will have an Install button
  5. Press install and the phone will try to verify the download but this will fail.
  6. You will have 2 buttons after the failed verification Close and Try again
  7. You can have some fun pressing the Try again button a few times only to watch it fail each time
  8. Move out of the range of the wireless and press try again and the verification will succeed – Crazy I know but you need to verify with 3G if the router is not configured for https
  9. At this point you can move back into the range of the wireless if you would like
  10. Installation will continue – about another 10-15 minutes
  11. At the end of the installation the phone will reboot
  12. You will be prompted for your Apple ID
  13. Enter your Apple ID if applicable if not then skip and ignore the iCloud warning
  14. Click Start using the iPhone

I tried this on 3 Apple iPhones, a 4 and a 4s. In all cases I experienced the iPhone Upgrade verification failure and it worked without a problem.
The wireless router that the iPhone started the upgrade with was not configured for https.
The verification process requires https to start. Moving out of the range of the wireless router allows
the verification to succeed through the 3G connection. Everything proceeds from there without a problem.

Or turn off the wireless.