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If you don't have a Time Machine backup you have to install the system manually.

If you don't have a Time Machine backup or another appropriate backup, you have to install the system manually.

• Quit Terminal
• Open 'Disk Utility' and check your newly created volume for errors
• Quit 'Disk Utility'
• Attach your external Time Machine backup drive or check this answer if you use NAS or another network share.
• Open 'Restore from Time Machine Backup'
• Choose the appropriate Time Machine backup and restore your system
• Reboot to your restored system.
• Unmount and detach your Time Machine backup drive
• Open 'Terminal' and enter 'diskutil list'
• Check if your 'Recovery HD' is listed.
• If your 'Recovery HD' is missing, usually reinstalling your current system with the latest available system installer (e.g. 'Install OS X Yosemite (10.10.5)' if Yosemite is currently installed) will recreate it without loosing any data.
• After reinstalling the system with the latest available system installer open App Store and install the latest security fixes.

• Quit Terminal
• Open 'Disk Utility' and check your newly created volume for errors
• Quit 'Disk Utility'

Restore Time Machine backup

• Attach your external Time Machine backup drive or check this answer if you use NAS or another network share.
• Open 'Restore from Time Machine Backup'
• Choose the appropriate Time Machine backup and restore your system
• Reboot to your restored system.
• Unmount and detach your Time Machine backup drive
• Open 'Terminal' and enter 'diskutil list'
• Check if your 'Recovery HD' is listed.
• If your 'Recovery HD' is missing, usually reinstalling your current system with the latest available system installer (e.g. 'Install OS X Yosemite (10.10.5)' if Yosemite is currently installed) will recreate it without loosing any data.
• After reinstalling the system with the latest available system installer open App Store and install the latest security fixes.

Restore from other sources

If you don't have a Time Machine backup you have to install the system manually.

• Booted to Internet Recovery Mode choose "Restore OS X" (or something similar). This will download and install the original OS X coming with your Mac.

• After installing OS X configure your Mac with your common user. Download and upgrade to the appropriate system (in your case probably 10.10.5). If you haven't "purchased" (it is or was free) OS X Yosemite with your Apple App Store account you are lost and you have to get it from somewhere else.

• After upgrading to OS X Yosemite attach your external drive and open the restore image.

• Copy all necessary files & folders in your old user folder to your new account. Avoid overwriting existing folders of your new account! Also check your old /Users/your_user_name/Library folder (which usually is invisible!).

• Download and reinstall all needed applications.

• To backup your main volume, boot to your main drive
• Attach an external drive. The drive has to be formatted as an HFS+ Journaled volume. Check this with Disk Utility. If necessary repartition the external drive. All data on the drive will be destroyed after repartitioning/formatting it. The drive has to have the size of the volume to back up (or more).
• Usually you will be asked if the external drive should serve as a Time Machine backup drive after plugging it in for the first time. If not, open Time Machine in the System Preferences and configure the external drive as Time Machine backup drive.

• To backup your main volume, boot to your main drive
• Attach an external drive. The drive (or at least one volume) has to be formatted as an HFS+ Journaled volume. Check this with Disk Utility. If necessary repartition the external drive. All data on the drive will be destroyed after repartitioning/formatting it. The drive (or at least one volume) has to have the size of the volume to back up (or more).
• Usually you will be asked if the external drive should serve as a Time Machine backup drive after plugging it in for the first time. If not, open Time Machine in the System Preferences and configure the external drive as Time Machine backup drive.