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**1. Homework Statement**

I'm inspecting a proof which can be found here: http://unafold.math.rpi.edu/Teaching/MATH-2800/Binary_Ops.pdf

My question regards the line: (a

_{1}...a

_{k-1})*(a

_{k}...a

_{n}) = (a

_{1}...a

_{k-1})*((a

_{k}...a

_{n-1})*a

_{n}) on the second page of the document.

Is this true (that is, the ability to group terms k through n on the RHS of the equality, as written) because "any bracketing of a

_{1}*a

_{2}*...a

_{n-1}equals the standard form..."? In other words, I am asking if the inductive hypothesis applies to any qualifying number of summands, regardless of whether they are the same terms as those explicitly expressed in the hypothesis.

Also, wouldn't this necessarily imply that 1 < k in the second inequality involving k? This may at first seem trivial, but I suppose you'd need fewer than n elements in the right-hand paranthetical expression for this step to hold, which it obviously wouldn't if k = 0.

**2. Homework Equations**

**3. The Attempt at a Solution**

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