It seems obvious that the home team's intent was that A10 is batting for A4, but if the substitution is unannounced, intent doesn't matter and the PH is always for the proper batter. So we have two possible series of events.
IF A10 WAS ANNOUNCED FOR A4: A10 is out of turn. After A10 is legitimized, the proper batter is A5, but A6 bats out of turn. After A6 is legitimized, the person A11 is batting for is A7. In the top of the next inning, A12 replaces A6.
IF A10 WAS UNANNOUNCED: A10 is batting for A6. The next batter is A7, but A6 bats instead; this is an illegal substitution, not BOOT. Because A6 is batting in the seventh spot in the order, the proper batter following A6 and the person A11 is batting for is designated hitter A8. In the top of the next inning, A11 moves to the field which terminates the DH, and A12 also enters the game, with A4 and illegal sub A6 exiting. A12 bats either fourth or seventh, with the pitcher taking the other spot.
Obviously if A10 was unannounced and the crew knew what was going on, they would prevent A6 from batting in A7's spot, so that (and A8 batting in the next inning) indicates A10 was announced.