Covariant derivative formula- question about change of index

Question

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How can you simply change superscript index j to k only for the 1st v & not for the 2nd v in the equation as shown in the diagram?

Answer 1

Each term needs to have the same uncontracted indices, at the same heights - in this case, a subscript $i$. That's because $X^i=Y^i$ means $X^{n_\min}=Y^{n_\min}$ etc., up to $X^{n_\max}=Y^{n_\max}$.

But the terms can differ in not only how many pairs of contracted indices they have, but also what they're called. The name in particular is irrelevant because $\sum_{j=n_\min}^{n_\max}X_j^j$ is the same thing as $\sum_{k=n_\min}^{n_\max}X_k^k$, namely $X_{n_\min}^{n_\min}+\cdots+X_{n_\max}^{n_\max}$.