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\begin{align}
&\color{#ff0000}{\int_{0}^{1}\fermi\pars{x}\,\dd x\int_{0}^{1}{1 \over \fermi\pars{x'}}\,\dd x'}
=
\half
\int_{0}^{1}\int_{0}^{1}\bracks{%
{\fermi\pars{x} \over \fermi\pars{x'}} + {\fermi\pars{x'} \over \fermi\pars{x}}}
\,\dd x\,\dd x'
\\[3mm]&=
\half
\int_{0}^{1}\int_{0}^{1}\braces{\vphantom{\Huge A^{A}}\,\bracks{\vphantom{\LARGE A^{A^{A}}}%
\root{\fermi\pars{x} \over \fermi\pars{x'}}
- \root{\fermi\pars{x'} \over \fermi\pars{x}}}^{\,2} + 2\root{\fermi\pars{x} \over \fermi\pars{x'}}\root{\fermi\pars{x'} \over \fermi\pars{x}}\,}
\,\dd x\,\dd x'
\\[3mm]&\color{#ff0000}{\geq \int_{0}^{1}\int_{0}^{1}\dd x\,\dd y = 1}
\end{align}
$$
\color{#0000ff}{\int_{0}^{1}\fermi\pars{x}\,\dd x\int_{0}^{1}{1 \ \fermi\pars{x'}}\,\dd x'}
\geq \color{#0000ff}{1}
$$
