Tommy, what we are going to do is, we can create a new Mac OS X Mountaion Lion
disk. We can use your Seagate external hard drive for that.
First, on your other Mac, goto AppStore and re-download Mountaion Lion. See here : http://osxdaily.com/2012/07/26/redownload-os-x-mountain-lion-mac-app-store/
After downloading Mountaion Lion, download the Lion Disk Maker application from here : http://liondiskmaker.com/
Then connect your external hard drive to your Mac and create a Mountaion Lion disk using the Lion Disk Maker
application. Make sure you choose your external hard drive in Lion Disk Maker and not the internal hard drive. This will erase your external hard drive and copy the Mountain Lion installation files to the external hard drive and make it bootable.
Once it is finished, connect the external hard drive to your Macbook Air, restart your Mac and immediately hold down the Option
key on your keyboard. Keep it held down and it will bring the Startup Manager screen. From there, select the external hard drive (it should have the name "Recovery HD") and press return
key. This will boot your Mac into Mountaion Lion installer. Then follow the onscreen instructions to reinstall the Mountaion Lion
FYI Command + R
while booting is Mountain Lion Internet Recovery. So you need to a high speed internet (broadband) to make it work.
Please let me know how it goes Tommy..