Hello,
We apologize for the issue you are facing with installing the PRO version of Rank Math after your purchase. Here are the steps you can follow to ensure a successful installation:
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Download Rank Math PRO: Log into your Rank Math account and download the Rank Math PRO plugin zip file.

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Install the Free Version: Make sure that the free version of Rank Math is installed and activated on your website. You can do this by going to WordPress Dashboard → Plugins → Add New, searching for Rank Math, and clicking Install Now followed by Activate.

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Upload the PRO Version: Navigate to WordPress Dashboard → Plugins → Add New, click on Upload Plugin, choose the Rank Math PRO zip file you downloaded, and click Install Now. After installation, click on the Activate now link.

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Activate Your License: To activate your Rank Math PRO license, go to WordPress Dashboard → Rank Math SEO → Dashboard → Help and click on Connect Now. Follow the prompts to connect your Rank Math account and activate your license.

If you have followed these steps and are still unable to install the PRO version, please reply to this ticket, and a human agent from the Rank Math Support team will assist you further. Thank you for raising this concern!
Sources:
Should I Delete the Free Version After Installing Rank Math PRO? » Rank Math
Should I Delete the Free Version After Installing Rank Math PRO? » Rank Math
How to Upgrade to Rank Math PRO » Rank Math
This is an automated reply from Rank Math Support Bot. If you need further assistance, please reply to this topic and a human support agent will reply to you shortly.
Hello,
Since we did not hear back from you for 15 days, we are assuming that you found the solution. We are closing this support ticket.
If you still need assistance or any other help, please feel free to open a new support ticket, and we will be more than happy to assist.
Thank you.