Case $K=N$: the solutions of $\sum_k(p^k)^2=1$, $0\le p^k\le1$, $\sum_kp^k=1$ are basis vectors

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Case $K=N$: the solutions of $\sum_k(p^k)^2=1$, $0\le p^k\le1$, $\sum_kp^k=1$ are basis vectors · Prove2Me