"Another Change-Scheme!"

In Poul Anderson's The Shield of Time (New York, 1991), the four Exaltationists still at large plan to change history by killing King Euthydemus of Bactria and his son Demetrius when they sally from their city of Bactra to attack the invaders led by Antiochus the Great. If one Exaltationist shoots the two Bactrian leaders on the battlefield, then he will thereby both generate and enter an alternative timeline but will he not also leave his three companions behind in the timeline regarded by the Time Patrol as "deleted"?

I think that the time-traveler who generates a new timeline must disappear from his original timeline. Such a disappearance by a time traveler is not mentioned in the series because what is referred to is the disappearance of the entire original timeline. However, I do not think that it makes sense to say that the world as we know it disappears at the moment when the time traveler changes history. Rather, he generates an entire alternative timeline in a second temporal dimension at right angles to his familiar temporal dimension. It is confusion between these two temporal dimensions that causes confusion when discussing time travel.

Time Patrol disinformation causes the Exaltationists to believe that another time criminal intends, a few years later, to usurp the throne of King Arsaces of Parthia, then to attack and kill Antiochus:

"'Another change-scheme!' Draganizu exclaimed.
"'Ours will nullify it and its operators,' Raor murmured." (p. 85)

How would the Exaltationists' change-scheme nullify a later one? I have argued that the world will not cease to exist at the moment of the Exaltationists' historical change. In the timeline guarded by the Time Patrol, neither Euthydemus nor Antiochus was assassinated. According to the logic of the Time Patrol series, a time traveler who does assassinate either monarch thereby generates and enters a new timeline in which an assassination did occur but what happens if both assassination attempts succeed at different moments of the same original timeline? I do not think that there is any way to answer this question.