The Second Continued Fraction Identity

Theorem (The Second Continued Fraction Identity)
\[[ a_1,a_2,..., a_{n-1}, a_n ] = [ a_1,a_2,...,a_n, a_{n-1}+ \frac{1}{a_n} ] \]
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Proof
\[\begin{equation} \begin{aligned} [ a_1,a_2,...,a_{n-1},a_n ] &= a_1+\frac{1}{a_2+\frac{1}{a_3+ \frac{1}{\ddots a_{n-1}+1/a_n}}} \\ &= a_1+\frac{1}{a_2+\frac{1}{a_3+ \frac{1}{\ddots (a_{n-1}+1/a_n)}}} \\ &= [ a_1,a_2,...,a_{n-1}+1/a_n ] \end{aligned} \end{equation}\]
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