The first prince felt utterly defeated since the nobles that had their attention on him were now trying to establish connections with the Supreme Mage. He couldn't even direct his hatred and anger toward the Supreme Mage since, unlike the Supreme Mage, he didn't have the power to destroy a kingdom.
The first prince was just a genius in magic, but not a prodigious monster like the Supreme Mage. The first prince was just a 5th Circle Mage while the Supreme Mage was an 8th Circle Mage.
No matter how much the first prince tried to use his intelligent mind to scheme against the Supreme Mage, he could never find a way to defeat the Supreme Mage. Even if he managed to instigate all of the people in the ballroom to go against the Supreme Mage, they wouldn't be able to defeat the Supreme Mage.