In chapter twelve of Anthem, by Ayn Rand, Prometheus realizes that "To be free, a man must be free of his brothers." What are some examples in Anthem that prove this statement correct?
Anthem, by Ayn Rand, is an allegorical statement about individuals in society, narrated by a character who is known for most of the story as Equality 7-2521 but who later, in a burst of individualism, changes his name to Prometheus. The quote you mention is followed by an equally important line to add emphasis:
“To be free, a man must be free of his brothers. That is freedom. This and nothing else.”
In the world of this novel, all men and women are brothers, and there is no distinction among them, except for those who administer and enforce equality (read sameness and lack of individuality).
A literal reading of the quote suggests that Equality 7-2521 can only find freedom when he is able to escape his brothers (his fellow men). In chapter two, Equality 7-2521 has discovered his tunnel and speaks for himself (though he uses the plural "we," as everyone is required to do) when he expresses how he feels when he can escape to it.
But here, in our tunnel, we feel it no longer. The air is pure under the ground. There is no odor of men. And these three hours give us strength for our hours above the ground.
It is ironic, but perhaps fitting, that the place he is able to escape the "odor [presence] of men" and energize himself for the torments of collective living, is a below-ground tunnel, a place which generally suggests confinement rather than freedom. In this case, not being with his brothers is true freedom.
In the first chapter, Equality 7-2521 says,
We are nothing. Mankind is all. By the grace of our brothers are we allowed our lives. We exist through, by and for our brothers who are the State.
So, a broader understanding of the Prometheus quote is that man must be free of the state (or the control of the state) in order to be free. This is a true statement for Equality 7-2521, for the state has confined and restricted him to a menial job and wants to punish him for thinking, creating, and experimenting. He has to escape from the state in order to be free, both literally and figuratively.
Prometheus speaks against the collective "we" (as in all men being seen as one collective) in his final grand speech:
The word "We" is as lime poured over men, which sets and hardens to stone, and crushes all beneath it, and that which is white and that which is black are lost equally in the grey of it. It is the word by which the depraved steal the virtue of the good, by which the weak steal the might of the strong, by which the fools steal the wisdom of the sages.
What is my joy if all hands, even the unclean, can reach into it? What is my wisdom, if even the fools can dictate to me? What is my freedom, if all creatures, even the botched and impotent, are my masters? What is my life, if I am but to bow, to agree and to obey?
But I am done with this creed of corruption.
I am done with the monster of "We," the word of serfdom, of plunder, of misery, falsehood and shame.
And now I see the face of god, and I raise this god over the earth, this god whom men have sought since men came into being, this god who will grant them joy and peace and pride.
This god, this one word:
Prometheus has disconnected himself from his brothers, from the collective, and is finally free.