似乎确实是正确的算式

请问以下推论正确吗?

&\sum_{r=l+1}^n (\sum_{i=l+1}^r a_i\times \sum_{i=l}^r b_i)-\sum_{r=l}^n (\sum_{i=l}^r a_i\times \sum_{i=l}^r b_i)\\ =&\sum_{r=l+1}^n\sum_{i=l+1}^r a_i b_i -\sum_{r=l}^n\sum_{i=l}^r a_i b_i \\ =&\sum_{r=l+1}^n\sum_{i=l+1}^r a_i b_i -(\sum_{r=l+1}^n\sum_{i=l}^r a_i b_i + \sum_{i=l}^l a_i b_i) \\ =&\sum_{r=l+1}^n\sum_{i=l+1}^r a_i b_i -(\sum_{r=l+1}^n\sum_{i=l}^r a_i b_i + a_l b_l) \\ =&\sum_{r=l+1}^n\sum_{i=l+1}^r a_i b_i -\sum_{r=l+1}^n\sum_{i=l}^r a_i b_i - a_l b_l \\ =&\sum_{r=l+1}^n(\sum_{i=l+1}^r a_i b_i -\sum_{i=l}^r a_i b_i) - a_l b_l \\ =&\sum_{r=l+1}^n(\sum_{i=l+1}^r a_i b_i -(\sum_{i=l+1}^r a_i b_i+\sum_{i=l}^l a_i b_i)) - a_l b_l \\ =&\sum_{r=l+1}^n(\sum_{i=l+1}^r a_i b_i -(\sum_{i=l+1}^r a_i b_i+a_l b_l)) - a_l b_l \\ =&\sum_{r=l+1}^n(\sum_{i=l+1}^r a_i b_i -\sum_{i=l+1}^r a_i b_i-a_l b_l) - a_l b_l \\ =&\sum_{r=l+1}^n(-a_l b_l) - a_l b_l \\ =&-a_l b_l(n-(l+1)+1) - a_l b_l \\ =&-a_l b_l(n-l) - a_l b_l \\ =&-a_l b_l(n-l+1) \\ \end{align*}

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