Here we have Mishnah 9 of the fourth chapter of Maseches Kiddushin, which deals with someone who appointed an agent to betroth and then went ahead and performed the betrothal himself - both in the case of a father betrothing his daughter and in the case of a woman betrothing herself.
The first case - the father and his agent:
The Mishnah opens: "Mi shenasan reshus lishlucho lekadesh es bito, vehalach hu vekidshah" - a man appointed an agent to betroth his daughter, and afterward he went and betrothed her himself. These are the laws that apply:
"Im shelo kadmu - kiddushav kiddushin" - if the father's own betrothal came first, that is the valid betrothal, and even if the agent betroths her afterward, his betrothal does not take effect.
"Im shel shelucho kadmu - kiddushav kiddushin" - if the agent betrothed her first, the agent's betrothal is the valid one, and whatever the father did afterward is not valid.
"Ve'im eino yadua - shneihem nosnin get" - when it is not known which of them betrothed her first, both potential husbands give her a get. "Ve'im ratzu - echad nosen get ve'echad kones" - if they wish, it is enough for one of them to give a get and for the other to marry her, since even if the one who gave the get is the true husband, once he has given the get the other may marry her.
The second case - the woman and her agent:
"Vechen ha'ishah shenasnah reshus lishluchah lekadshah, vehalchah vekidshah es atzmah" - here the Mishnah speaks of the woman herself rather than her father, and the law is identical to the law of the father:
"Im shelah kadmu - kiddusheha kiddushin" - if the woman's own betrothal came first, hers is the valid one and not the agent's.
"Im shel sheluchah kadmu - kiddushav kiddushin" - if the agent's betrothal came first, his is the valid one, while her own betrothal, which came afterward, does not take effect.
"Im eino yadua - shneihem nosnin lah get" - when there is a doubt as to who betrothed her first, both give her a get. "Ve'im ratzu - echad nosen lah get ve'echad kones" - if the husbands wish, one gives her a get and the other marries her.
Why were both cases needed?
The Gemara explains why the Mishnah taught two cases that are almost entirely alike - one concerning a father and one concerning the woman herself. The underlying principle is that since the one who appointed the agent went and betrothed on his own, it is clear that he wants his own betrothal to take effect if it comes first, and by doing so he has canceled the agent's appointment to betroth. Yet with each case on its own there would have been room to say otherwise:
Had the Mishnah taught only the law of the father: we would have said that a father is well versed in matters of lineage and the like, so he certainly knows what he is doing, and when he betrothed her first his betrothal is valid and the agent's appointment is canceled. But regarding a woman we would have suspected that she is not confident in herself, and that she might say to herself: I do not know for certain what is right here - and therefore she does not rely on her own betrothal, and she did not fully cancel the agent's appointment, but rather wanted the agent's betrothal to take effect. The Mishnah therefore taught the second part as well: even with a woman, if she betrothed herself first, her betrothal is valid and not the agent's.
Had the Mishnah taught only the law of the woman: we would have said that a father, since this is not his own marriage but his daughter's, is not so committed to the act of betrothal that he performed and did not intend that it definitely take effect, but rather wanted to see whether what the agent would do might appeal to him - and so he did not cancel the agent's appointment. The Mishnah therefore taught the law of the father as well.
The Mishnah thus teaches in both cases alike: the betrothal that came first in time - whether performed by the father or by the woman herself - is the valid betrothal, and the agent's appointment is canceled.
In summary: we have learned that one who appointed an agent, whether a father or a woman, and then betrothed on his own, thereby cancels the agency, and the betrothal that came first in time is the valid one. When there is a doubt as to which came first, both give a get, and if they wish, one gives a get and the other marries her. Both cases were taught in the Mishnah in order to rule out the reasoning that the father or the woman did not intend to cancel the agent's appointment.