Solved: 407 Proxy Authentication Required (Feisty)

In both :
1. System->Preferences->Network Proxy-> Set : Direct Internet Connection.
2. System->Administration->Synaptic Package Manager->Settings->Preferences->Network-> Set : Direct Internet Connection.

Then create a filename in:
/etc/apt/apt.conf

add the following lines:

Acquire::http::Proxy "http://username:password@proxy:port";
Acquire::ftp::Proxy "ftp://username:password@proxy:port";

and save the file.

In the terminal run:

sudo apt-get update

Both apt and synaptic should work now. (You might need to restart your networking or your PC)