Firstly, i'd like the state that this works ONLY on the iPhone 4s - it does not work on any other iOS device. Don't bother trying or it'll get bricked.

To start off, you must have a jailbroken iPhone 4s, running iOS 7.x or 8.x

You also need the following downloads:

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Now, for the tutorial.

1. Uninstall your current version of iTunes and all apple related software.
2. Install iTunes 12.0.1
3. Install Win32 OpenSSL
4. On your iPhone open Cydia and install OpenSSH
5. Extract kDFU.zip and run the exe
6. Enter your iPhone's IP address and press start
7. Wait for it to finish, your iPhone 4s will have a black screen - it will be in kDFU mode
8. extract downgrade.zip to your desktop
9. put fistmedaddy.ipsw into the downgrade folder on your desktop
10. Open command prompt as admin
11. In command prompt type "cd C:\Users\YOURUSER\Desktop\downgrade" replacing YOURUSER with your PC username
12. In command prompt type "idevicerestore -e fistmedaddy.ipsw"
13. If Command Prompt is stuck at "Extracting filesystem from IPSW" don't worry, it takes a while to extract.
14. Once done, your iPhone should boot to the Activation screen. Pass all that and congratulation, your iPhone 4s is now on 6.1.3!
15. You can use [ Register or Signin to view external links. ] to jailbreak again