The narrator starts off by correcting Hillary's account and telling his own version of the story - and then, he supplements the story with a previously unknown detail: who made it first to the top.
Right from the beginning, he says that Hillary's account was not completely true. Hillary had claimed that the narrator ran out of breath many times. However, according to the narrator, both of them were equally hardy, and both of them equally supported each other.
Then, he goes on to explain how the two of them were partners, rather than the one who leads and the one who follows. Neither of them would have made it without the other. Mountaineers know this, he says, but common people only wish to know who it was that first made it to the top. And then, he uncovers this detail: it was Hillary who first stepped there. Thereby, the narrator reinforces his position, as the person who tells the truth, even when it is unfavorable to him.
