That screen indicates that the firmware/software of your iPhone is corrupted because you have interrupted the update process. You shouldn't have done that. You have to force-restore the iPhone to factory settings through iTunes application (from computer) by putting it to DFU (Device Firmware Upgrade) mode, which will erase everything from it. Your iPhone is asking you to do this. If your work computer doesn't allow you to install apps, then you have to do this from a personal computer.
If you had backed up your iPhone to iCloud, then you can restore all your data back after restoring your iPhone from iCloud. You can also redownload all your past purchases for free using the same account.
Follow the steps below to restore your iPhone from your computer.
First download and install latest iTunes on your computer from here : https://www.apple.com/itunes/download/
Then connect the iPhone to your computer with the USB Cable.
Open iTunes application on your computer. iTunes will ask you to update or restore your iPhone. Try Update first. If that didn't work, then repeat the steps again and this time click Restore.
Remember, it may take a while to restore the iPhone (the iOS firmware has to be downloaded, you will see the Download button - a downward arrow button with spinning circle around it, at the top right corner of iTunes screen. Click it to see the time remaining)
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