Hmm. I agree with you but with a caveat; a belief in fate is more fundamental than a belief in fate guided by a divine entity. That's my only sticking point, but I could easily argue that they are the same thing, but for me, they are not.
My assumption is that if you were to ask Nietzsche if he believed in fate, he would say yes (obviously, this is apparent). Now ask if believed a divine entity was the direct cause of this, he would say no.
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