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Import using Terminal Commands
Terminal commands are a more powerful way to interact with GPG. They can be used on any of the operating systems mentioned in this post. I will demonstrate using an VM running Ubuntu 18.04 Desktop. You can run these same commands in Windows using PowerShell, and in Mac OS using the Terminal app.
Import ThomasV's public key:
gpg --import /<path>/<to>/<file>/<location>/ThomasV.asc
Example:
gpg --import ~/Downloads/ThomasV.asc
The response should look like this:

Refresh your keyring:
gpg -k
You should now see ThomasV's key in your keyring, the entry should look like this:
If you choose to, you can now certify the key. You must have created or imported a secret key. If you created one, it will automatically be set as the default and trust level will be set to ultimate. If you imported an existing secret key, you'll have to set the trust level and defaults manually. Refer to the GnuPG documentation for instructions.
gpg --sign-key 6694D8DE7BE8EE5631BED9502BD5824B7F9470E6
Optionally, if you have multiple secret keys, or don't have one set as a default, use this command to certify ThomasV's public key:
gpg -u <yourfingerprint> --sign-key 6694D8DE7BE8EE5631BED9502BD5824B7F9470E6
Select y and press enter at the two following prompts. You'll be prompted for the GPG password that you set when creating your key pair. ThomasV's key trust level will be set to "full."
Check the trust level of the public key by refressing the keyring:
gpg -k
The results for ThomasVs key should look like this:

Download and Verify Electrum
.
Download and Verification
Browse to the official Electrum website, and then to the downloads page. Make sure it's the official site. I know, it sounds easy, right? But this is the riskiest part of the whole process. Scammers want you to mistake their site for the real one, so they do everything they can to lure you into their trap.
Download the package or binary for your system, and save it somewhere you'll remember. Download the corresponding signature file, and save it in the same location.
Windows users; pay close attention to the file extension. Windows wants to save it as a .txt file, but we want it saved as .asc.
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Verify on Windows
In Kleaopatra, click on the "Decrypt/Verify" button, and browse to the location of the .exe and .asc files you saved. Select the .asc file, and click "Open."

The software will check the integrity of the .exe file and compare it to the signature file. If the signature matches the .exe file you'll see a window like this pop up:

Conclusion
.
Verify on Mac OS
Verification on a Mac is easy, just open a Finder window, navigate to the location where you save the Electrum .dmg file, and the .asc signature file, and double click the signature file.

Mac GPG will launch the verification tool, and compare the .dmg file to the signature file. Once the verificaiton tool has competed it's diagnostic it'll pop up a window like this:

Conclusion
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Verify using Terminal Commands
To verify the downloaded AppImage enter the following command:
gpg --verify /<path>/<to>/<file>/<location>/<filename>.AppImage.asc
Example:
gpg --verify ~/Downloads/electrum-3.3.8-x86_64.AppImage.asc
If you have certified ThomasV's public key the result should look like this:
gpg: assuming signed data in '/home/<usernam>/Downloads/electrum-3.3.8-x86_64.AppImage'
gpg: Signature made Thu 11 Jul 2019 07:26:15 AM PDT
gpg: using RSA key 6694D8DE7BE8EE5631BED9502BD5824B7F9470E6
gpg: Good signature from "Thomas Voegtlin (
https://electrum.org) <
thomasv@electrum.org>" [full]
gpg: aka "ThomasV <
thomasv1@gmx.de>" [full]
gpg: aka "Thomas Voegtlin <
thomasv1@gmx.de>" [full]

Conclusion
.
Conclusion
That's it, you did it! Now you have the tools and the skill to verify that the Electrum installation files you download are authentic, and not the work of some malicious scammer.
Stay safe, and happy bicoining!