Returning to our site, we enter the old URL which we just mapped, and see that the new mapping isn't working... why is that?

The reason is that the old page still exists on the server. When the server receives a request for a page, it first looks for a file matching that page name. If it finds it, it will send back that file to the browser. Otherwise, it will try something else, like looking in the rules from the .htaccess file.

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