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Well, regardless of who you remove, you can't change the original timeline. That original timeline will just keep trucking along as usual. What WILL happen is that a new timeline will branch off; one in which said person no longer exists. But you would never see it unless you travelled back in time and affected the change personally; meaning you're now in that timeline; an active part of it, and thus never able to return to your original. If you travel forward again, you're still in the new timeline. If you travel back, you create yet another new timeline; once in which the wave function collapsed to show that you travelled back in time.
It's a wave function collapse. We live in a universe where that wave function collapsed to say said person existed. If we blink them out of existence, we're essentially re-opening the box, which creates a new tangent. But that original collapse still exists. It happened; it's immutable.
Re-rolling a dice doesn't change the fact that it's already been rolled before.
Weirdly enough, the best example of that is ironically the season three episode of Community called "Remedial Chaos Theory"
This power allows you to change the original timeline.