Is what the worshiper did—completing the Tashahhud after forgetting a prostration in the last rak‘ah, then performing it, then saying the Taslim, then performing the prostration of forgetfulness—correct, or must he repeat the prayer?

The correct approach was to repeat the Tashahhud after performing the prostration, because it occurred after something that was omitted and thus was not counted. The Fajr prayer must be repeated on the basis that the final Tashahhud is a pillar of the prayer. Some scholars have said: The prostration of forgetfulness (sujud al-sahw) is sufficient for omitting the final Tashahhud. However, repeating the prayer is better and closer to piety, as it avoids the differing opinions of the scholars.