Well yes, showjumping is dressage with some obstacles along the way.
In puissance you have 1-3 fences overall. 2 oh those being the warm up fences. So you essentially have one real jumping effort in the whole arena. If your lead up to that fence is crooked or zig zagging everywhere etc etc you are not going to stand a very good chance of clearing the fence. The horse also needs to be round and collected in order to get the best bouncy canter. So yep... Dressage ;)
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The first round of a Puissance has six or seven large fences. All fences
count in the first couple of rounds. Each round a fence is removed and the rest made higher and wider until there is a smaller first fence, which I believe now still counts if it is knocked down, the triple bar and the wall.
Some riders have a horse that 'specialises' in Puissance but no order does just those classes. You often find that after such a class the rider will take the horse in a speed class where the fences are lower, so it is not worried about the stress of large heights.
As for dressage, agreed that show jumpers are well schooled, but in a different way to pure dressage horses. Most are ridden frequently in draw reins, they can lengthen and shorten a stride easily from the rider's seat and the rider can put any of his horse's feet on a dime!
Going into a big wall the horse is kept on a shorter stride until three or four strides away from the fence whereby most riders allow them to move faster and longer, though not stretched out.
Not a class I particularly like!