1. He authorizes one of them.
2. He authorizes both of them.
3. He authorizes neither of them.

If he authorizes one of them, then it is valid, whether it is later or earlier. If he authorizes both of them, or authorizes neither of them, then the second is invalid, according to what the author’s words imply.

What is meant by “the second” is the one whose opening takbir occurs after that of the other, even if the other was established earlier. But how can this be known?

The answer is that in earlier times, knowing which of the two preceded the other in the opening takbir might have been difficult. In the present time, however, it may be easy to know which of them preceded the other by means of a loudspeaker: if we hear the imam of the first say, “Allah is the Greatest,” and then the imam of the second says immediately afterwards, “Allah is the Greatest,” we say to the second: “Your prayer is invalid,” and to the first: “Your prayer is valid.” This is because, once the first preceded the other in the opening takbir, the obligatory duty became attached to it, since it came first. According to the madhhab, the prayer is deemed to have begun with the opening takbir. Thus, once it preceded the other with the opening takbir, the obligation became attached to it, and it became the obligatory prayer; the second is invalid.

Some scholars said: What counts is chronological precedence. Thus, the one established first takes precedence, because the second is an intrusion upon the first. It therefore resembles the Mosque of Harm, which the hypocrites built beside the Mosque of Quba. Allah said to His Prophet: “Do not stand in it ever.” [Al-Tawbah 9:108]