after I tried the system restore in itunes on the iphone 4 I get an error in activation.

This usually happens if the phone has been Hacked / Jailbroken / Modified...

Is this the case...?

As your Posts to other threads have indicated... Your Device is indeed Jailbroken.

The discussion of Jailbroken Devices is against the Terms of Use of this Forum.

As a consequence there is No support from Apple or this Forum.

Sorry... But...

You will need to look elsewhere.

Unauthorized modification of iOS

http://support.apple.com/kb/HT3743