The invisible leash
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Actually, for a horse, it's called a "lead rope," not a "leash."
This probably has little to do with the horse thinking it's wearing a halter and lead, and more to do with how the horse was trained to follow.
I would do this with Simon all the time. I'd bring him into the arena, and stand to his left. With no lead rope, I'd say, "Walk with me," and he'd follow along just like the video. When I'd stop, he'd stop. When I'd take steps backward, so would he.
