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\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\natnums}{\mathbb{N}}
\newcommand{\cnums}{\mathbb{C}}
\newcommand{\znums}{\mathbb{Z}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
\newcommand{\rb}{\right]}
\newcommand{\supth}{^{\text{th}}}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}[proposition]{Definition}
\newcommand{\kf}{\mathbb{K}}
\newcommand{\KP}{{\mathbb{K}\mathrm{P}}}
\newcommand{\RP}{{\reals\mathrm{P}}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\nzkn}{\kf^{n+1}\backslash \{0\}}
\newcommand{\PSL}{{\mathrm{PSL}}}
\newcommand{\SL}{{\mathrm{SL}}}
\newcommand{\PGL}{{\mathrm{PGL}}}
\newcommand{\GL}{{\mathrm{GL}}}
\newcommand{\GammaL}{\operatorname{\Gamma L}}
\newcommand{\PGammaL}{{\operatorname{P\Gamma L}}}
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