(Mishnah - Beis Shamai): One may divorce his wife only if he found in her Ervah (immorality) - "Ki Matza Bah Ervas Davar";

Beis Hillel say, even if she burned the food [he may divorce her] - "Ki Matza Bah Ervas Davar (Ervah or a Davar (matter))";

R. Akiva says, even if he found a prettier woman - "if she does not find grace in his eyes..."

(Gemara) Question (against Beis Shamai - R. Shila of Kefar Timrasa): "Lo Yuchal Ba'alah ha'Rishon Asher Shalchah Lashuv Lekachtah" - what does this teach?

If it is to forbid her to him - [this cannot be, for] she is already forbidden (he found in her Ervah)!

Answer: Rather, it forbids her with a Lav. (Normally, only an Aseh forbids an Eshes Ish who was Mezanah.)

Rebuttal (and Answer #2 - Beraisa - Beis Shamai) Question: I would know that one may divorce only due to Ervah (Zenus). What is the source to divorce if her head is exposed, the sides [of her garment] are torn, or her arms come out [of her garment]?

Answer: "Ki Matza Bah Ervas Davar (Ervah or a Davar (matter)." In the latter case, she is permitted to him, but he may divorce her. If she married someone else, and was widowed or divorced, "Lo Yuchal..." forbids her to her first husband. We explained this like SEFER NIR.)

Question: How do [Beis Hillel] establish [the verse of] Beis Shamai? (Why does it say "Ervas"? Had it said only "Davar", this would imply even a blunder of burning the food, and all the more so a matter of Ervah! - PF)

Answer: We should not say that one who was divorced due to Ervah is forbidden (to marry anyone else), and [one who was divorced] due to another Davar is permitted. (It wrote both to teach that both may remarry. We explained this like PNEI MOSHE.)

It says "veha'Davah b'Nidasah veha'Zav Es Zovo" - early Chachamim used to say, she will be in her [status of] Nidah (i.e. distanced); she will not paint her eyes or apply mudpacks until she comes in water (immerses);

R. Akiva: Do you bring a proof from there?! If you say so, she will even become repulsive, and he will seek to divorce her!

The [early] Chachamim hold like Beis Shamai (he cannot divorce her even if she becomes repulsive). R. Akiva holds like Beis Hillel. (RASHBA asks, we should say that R. Akiva holds like he taught in our Mishnah! MAR'EH HA'PANIM answers that we teach that becoming repulsive is a misdeed like burning his food; Beis Hillel permit divorcing due to this. MISHNEH L'MELECH (Hilchos Gerushin 10:21) answers that that if she becomes repulsive, he will seek a Heter divorce her, e.g. burning his food. Had she not become repulsive, perhaps he would not divorce her due to this.)