The Searchers (1956)

Continuity mistake: In the scene where John Wayne is looking down at the arrow that Look has left, his moccasin is all brown. In the next shot, his moccasin is partially opened on the side, revealing his sock.

Laurie Jorgensen: Ethan rode on an hour ago. Now Martin I don't know what you can do about finding Debbie that he can't. He'll find her now, Martin. Honest he will.
Martin: That's what I'm afraid Of Laurie, him finding her. Oh I've seen his eyes at the very word "Comanche." I've seen him take his knife and - - Never mind. He's a man that can go crazy wild and I intend to be there to stop him in case he does.

Question: Is the buckskin horse Martin is riding when they go on the search for the girls the same horse that he rode to death in the scene before?