That Joff, Tommen, and Myrcella are Baratheon's and not incest-born Lannisters?

The don't have to prove that. Cersei was married to the king, and during their marriage had 3 children who were accepted by the king as his own. And after his death, they killed all of his bastards that could presumably show that all seeds of Baratheon have dark hair (and undermine their own claim potentially). They don't have to prove shit.