The ruling on one who prayed while in a state of ritual impurity and then realized it: is he required to repeat the prayer?

The Shaykh : So we return now to this man who mentioned that he realized he was in a state of ritual impurity during the prayer. He has a role model and an example in the Messenger of Allah (peace and blessings be upon him). This is conceivable if he is in his home, meaning the ablution area or the bathroom is close to him; otherwise, in most cases, this is not feasible. But does this ruling—that the prayer of one who forgot a pillar or a condition of the validity of the prayer, which is ritual purity, remains valid—does this ruling continue even if he has finished the prayer? The answer is no. That is, one who remembers his ritual purity after finishing the prayer does not have the same ruling as one who remembers it while in the middle of the prayer. So, the one who remembers that he is in a state of ritual impurity during the prayer and is able to perform ablution or take a full ritual bath, he returns and continues his prayer. As for the one who remembers that he is in a state of ritual impurity after he has finished the prayer, he must, in this case, restart the prayer from the beginning to the end. What is the evidence? If it were not for this evidence, we would have extended the previous ruling even to the one who prayed and said the closing salam while in a state of ritual impurity. But the following evidence is what prevented us from extending the ruling even to this one who remembered his state of ritual impurity after the salam. What is it? It is in Sahih al-Bukhari: This is the meaning of his words, because I do not have the exact wording memorized. And considering that he was first among the Rightly Guided Caliphs, whom the Messenger of Allah (peace and blessings be upon him) commanded us to follow their way along with his way in the famous hadith of al-'Irbad, in which he says: So, by looking at the fact that he was among these Rightly Guided Caliphs first, and considering that he said this statement in front of the attending Companions, and no one refuted him with the previous evidence in the first scenario, we said that it is necessary to distinguish between the one who remembers during the prayer, who takes a full ritual bath or performs ablution and continues, and the one who remembers after the prayer, who must restart the prayer after purifying himself. This is what needed to be explained regarding that question. Please proceed.