$\DeclareMathOperator*\K{K}$ Let $\K$ a operator of continued fraction. Show that: $$1+\sum_{k=1}^{n}(-1)^{k}\varphi_4\cdots \varphi_{2k+2}=\frac{1}{1+\displaystyle\K_{k=2}^{n+1}\frac{\varphi_{2k}}{1-\varphi_{2k}}}$$
What's a name of this little theorem? This theorem is true?