Yes, so long as you're logged into Google Play with the same account as was used to purchase the app and, naturally, you're attempting to download and install the correct app - if you try to install one that you haven't already purchased you'll correctly be prompted to pay.
Our Android apps all now qualify for Google Play Family Library.
You can check the account being used by Google Play by selecting the circular profile icon in the upper right corner, shown highlighted in red on the following screenshot:
The active account is then shown in the area highlighted in red on the following screenshot:
If the wrong account is selected, touch the down arrow to the right of the current account name and select the correct account.
If the correct account isn't shown, connect to another account as follows:
Select "Add another account" and then continue to enter the requested details to complete this process.
Once the account has been added, return to the Google Play app and select it before attempting to install the app again.