I am self-studying Spivak's Calculus.
In Chapter 1, Question 24, $a_1 + \ldots + a_n$ is defined as $a_1 + (a_2 + (a_3 + \ldots + (a_{n-2} + (a_{n-1} + a_n))) \ldots )$.
In part (a), we are asked to proof that $(a_1 + \ldots + a_k) + a_{k+1} = a_1 + \ldots + a_{k+1}$. The hint given was to use induction.
In the answers, it is written that:
\begin{align} (a_1 + \ldots + a_{k+1}) + a_{k+2} &= [(a_1 + \ldots + a_{k}) + a_{k+1}] + a_{k+2} \\ &= (a_1 + \ldots + a_{k}) + (a_{k+1} + a_{k+2}) \\ &= a_1 + \ldots + a_{k} + (a_{k+1} + a_{k+2}) \\ &= a_1 \ldots + a_{k+2} \end{align}
For the first and third equality, it is true because the equation holds for $k$. For the second equality, it is true because $a + (b + c) = (a + b) + c$. However, for the last equality, it is written that it is true by the definition of $a_1 + \ldots + a_{k+2}$.
I do not understand how the last inequality is derived from the definition. Any help will be appreciated.