\documentclass{article}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
\newcommand\unicdbdjdcd{\Rightarrow}
\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{t23\_absvalue (TMcpJ41CJUPfpA15RfHPBaX4jzGvsjbTG3m)}
\maketitle
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_absvalue : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k1\_real\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k18\_complex1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k6\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v2\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ np\_\_0 : {\iota}$ be given.
Let $ k3\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_xcmplx\_0 : {\iota}$ be given.
Let $ k4\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((r1\_xxreal\_0~(k1\_real\_1~(k18\_complex1~\\
X0))~X0){\unicdcdcdhd}(r1\_xxreal\_0~X0~(k18\_complex1~X0)))\end{array}
\label{hyp:ehedpfbgcgdhghbgmgfhfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xreal\_0~X2){\unicdbdjdcd}(\preoh (r1\_xxreal\_0~k6\_numbers~X0){\unicdcdcdhd}((\preoh r1\_xxreal\_0~\\
X2~X1){\unicdcdcdhd}(r1\_xxreal\_0~(k2\_xcmplx\_0~X0~X2)~X1)))))\end{array}
\label{hyp:ehedadpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xreal\_0~X2){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~X0~k6\_numbers){\unicdcdcdhd}((r1\_xxreal\_0~\\
X1~X2){\unicdcdcdhd}(r1\_xxreal\_0~(k2\_xcmplx\_0~X0~X2)~X1)))))\end{array}
\label{hyp:ehddjdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xreal\_0~X2){\unicdbdjdcd}(((r1\_xxreal\_0~k6\_numbers~X0){\unicdcdcdhd}(r1\_xxreal\_0~\\
X1~X2)){\unicdbdjdcd}(r1\_xxreal\_0~X1~(k2\_xcmplx\_0~X0~X2)))))\end{array}
\label{hyp:ehddidpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xreal\_0~X2){\unicdbdjdcd}(((r1\_xxreal\_0~X0~k6\_numbers){\unicdcdcdhd}(r1\_xxreal\_0~\\
X1~X2)){\unicdbdjdcd}(r1\_xxreal\_0~(k2\_xcmplx\_0~X1~X0)~X2))))\end{array}
\label{hyp:ehddfdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(\preoh (r1\_xxreal\_0~\\
k6\_numbers~X0){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X1~k6\_numbers){\unicdcdcdhd}(r1\_xxreal\_0~(\\
k2\_xcmplx\_0~X0~X1)~k6\_numbers))))\end{array}
\label{hyp:ehddedpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}((r1\_xxreal\_0~\\
X0~k6\_numbers){\unicdbdjdcd}(r1\_xxreal\_0~(k2\_xcmplx\_0~X0~X1)~X1)))\end{array}
\label{hyp:ehddcdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((X0=k6\_numbers){\unicdbdjded}(k18\_complex1~\\
X0=k6\_numbers))\end{array}
\label{hyp:ehcdpfbgcgdhghbgmgfhfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xcmplx\_0~X1){\unicdbdjdcd}(X0=\\
k6\_xcmplx\_0~(k2\_xcmplx\_0~X0~X1)~X1))\end{array}
\label{hyp:ehcdgdpfihdgngahmgihpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}(\preoh (k1\_absvalue~X0=k1\_real\_1~np\_\_1){\unicdcdcdhd}\\
(r1\_xxreal\_0~k6\_numbers~X0))\end{array}
\label{hyp:ehbdfdpfbgcgdhghbgmgfhfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}(\preoh (k1\_absvalue~X0=np\_\_1){\unicdcdcdhd}(r1\_xxreal\_0~\\
X0~k6\_numbers))\end{array}
\label{hyp:ehbdedpfbgcgdhghbgmgfhfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((v2\_xxreal\_0~np\_\_2){\unicdcdcdhd}(m2\_subset\_1~np\_\_2~k1\_numbers~k5\_numbers)){\unicdcdcdhd}\\
((m1\_subset\_1~np\_\_2~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_2~k1\_numbers))\end{array}
\label{hyp:dhahdgcdpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((v2\_xxreal\_0~np\_\_1){\unicdcdcdhd}(m2\_subset\_1~np\_\_1~k1\_numbers~k5\_numbers)){\unicdcdcdhd}\\
((m1\_subset\_1~np\_\_1~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_1~k1\_numbers))\end{array}
\label{hyp:dhahdgbdpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(m2\_subset\_1~np\_\_0~k1\_numbers~k5\_numbers){\unicdcdcdhd}((m1\_subset\_1~np\_\_0~\\
k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_0~k1\_numbers))\end{array}
\label{hyp:dhahdgadpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
{{{{k3\_xcmplx\_0}~{k1\_xcmplx\_0}}~{k1\_xcmplx\_0}}={{k4\_xcmplx\_0}~{np\_\_1}}}\label{hyp:chbhcffgbgmgnefhmgehpfpflgddpfihdgngahmgihpfadpfpfdgchbdchadpfdgchbdchadpfchngbd}
\end{equation}
Assume the following.
\begin{equation}
{{{{k6\_xcmplx\_0}~{np\_\_1}}~{np\_\_1}}={np\_\_0}}\label{hyp:chbhcffgbgmgeejgggggpfpflggdpfihdgngahmgihpfadpfpfchbdpfchbdpfchad}
\end{equation}
Assume the following.
\begin{equation}
{{{{k2\_xcmplx\_0}~{({{k4\_xcmplx\_0}~{np\_\_1}})}}~{np\_\_1}}={np\_\_0}}\label{hyp:chbhcffgbgmgbeegegpfpflgcdpfihdgngahmgihpfadpfpfchngbdpfchbdpfchad}
\end{equation}
Assume the following.
\begin{equation}
{{{{k2\_xcmplx\_0}~{np\_\_1}}~{({{k4\_xcmplx\_0}~{np\_\_1}})}}={np\_\_0}}\label{hyp:chbhcffgbgmgbeegegpfpflgcdpfihdgngahmgihpfadpfpfchbdpfchngbdpfchad}
\end{equation}
Assume the following.
\begin{equation}
{{{{k2\_xcmplx\_0}~{np\_\_1}}~{np\_\_1}}={np\_\_2}}\label{hyp:chbhcffgbgmgbeegegpfpflgcdpfihdgngahmgihpfadpfpfchbdpfchbdpfchcd}
\end{equation}
Assume the following.
\begin{equation}
{{{{k2\_xcmplx\_0}~{np\_\_1}}~{np\_\_0}}={np\_\_1}}\label{hyp:chbhcffgbgmgbeegegpfpflgcdpfihdgngahmgihpfadpfpfchbdpfchadpfchbd}
\end{equation}
Assume the following.
\begin{equation}
{{{{k2\_xcmplx\_0}~{np\_\_0}}~{np\_\_1}}={np\_\_1}}\label{hyp:chbhcffgbgmgbeegegpfpflgcdpfihdgngahmgihpfadpfpfchadpfchbdpfchbd}
\end{equation}
Assume the following.
\begin{equation}
{{{r1\_xxreal\_0}~{({{k4\_xcmplx\_0}~{np\_\_1}})}}~{np\_\_1}}\label{hyp:chbhmefgdhdhpechfebhfhbgmgpfpfchbdpfihihchfgbgmgpfadpfpfchngbdpfchbd}
\end{equation}
Assume the following.
\begin{equation}
{{{r1\_xxreal\_0}~{np\_\_1}}~{np\_\_1}}\label{hyp:chbhmefgdhdhpechfebhfhbgmgpfpfchbdpfihihchfgbgmgpfadpfpfchbdpfchbd}
\end{equation}
Assume the following.
\begin{equation}
{\preoh{{{{r1\_xxreal\_0}~{np\_\_1}}~{np\_\_0}}}}\label{hyp:chbhmefgdhdhpechfebhfhbgmgpfpfchbdpfihihchfgbgmgpfadpfpfchbdpfchad}
\end{equation}
Assume the following.
\begin{equation}
{{{r1\_xxreal\_0}~{np\_\_0}}~{np\_\_1}}\label{hyp:chbhmefgdhdhpechfebhfhbgmgpfpfchbdpfihihchfgbgmgpfadpfpfchadpfchbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(m1\_subset\_1~X1~k1\_numbers)){\unicdbdjdcd}\\
(k5\_real\_1~X0~X1=k6\_xcmplx\_0~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{{k1\_real\_1}~{X0}}={{k4\_xcmplx\_0}~{X0}}})}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}(k1\_absvalue~(k1\_absvalue~X0)=k1\_absvalue~\\
X0)\end{array}
\label{hyp:ahchpgkgfgdgehjgghjgehjhpflgbdpfbgcgdhghbgmgfhfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xreal\_0~X2){\unicdbdjdcd}((r1\_xxreal\_0~X0~(k2\_xcmplx\_0~X1~X2)){\unicdbdjdcd}(r1\_xxreal\_0~\\
(k6\_xcmplx\_0~X0~X1)~X2))))\end{array}
\label{hyp:mgcdjdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{{{k2\_xcmplx\_0}~{np\_\_1}}~{({{k4\_xcmplx\_0}~{np\_\_1}})}}={k6\_numbers}}\label{hyp:mgbdddpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdbdjdcd}(v1\_xreal\_0~\\
(k2\_xcmplx\_0~X0~X1))\end{array}
\label{hyp:ggdgfdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((v1\_xcmplx\_0~(k4\_xcmplx\_0~X0)){\unicdcdcdhd}\\
(v1\_xreal\_0~(k4\_xcmplx\_0~X0)))\end{array}
\label{hyp:ggdgddpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xreal\_0}~{X0}})}{\unicdbdjdcd}{({{v1\_xreal\_0}~{({{k1\_absvalue}~{X0}})}})}}}\label{hyp:egehpflgbdpfbgcgdhghbgmgfhfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}(((\preoh r1\_xxreal\_0~X0~k6\_numbers){\unicdbdjdcd}(\\
k1\_absvalue~X0=np\_\_1)){\unicdcdcdhd}(((\preoh r1\_xxreal\_0~k6\_numbers~X0){\unicdbdjdcd}(k1\_absvalue~\\
X0=k1\_real\_1~np\_\_1)){\unicdcdcdhd}(((r1\_xxreal\_0~X0~k6\_numbers){\unicdcdcdhd}(r1\_xxreal\_0~\\
k6\_numbers~X0)){\unicdbdjdcd}(k1\_absvalue~X0=k6\_numbers))))\end{array}
\label{hyp:egcdpfbgcgdhghbgmgfhfg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xreal\_0}~{X0}})}}}\label{hyp:dgdgbdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xcmplx\_0}~{X0}})}}}\label{hyp:dgdgbdpfihdgngahmgihpfad}
\end{equation}
\begin{theorem}\label{thm:ehcdddpfbgcgdhghbgmgfhfg}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(r1\_xxreal\_0~\\
(k5\_real\_1~(k2\_xcmplx\_0~(k1\_absvalue~X0)~(k1\_absvalue~X1))~np\_\_1)~\\
(k1\_absvalue~(k2\_xcmplx\_0~X0~X1))))\end{array}$$
\end{theorem}
\end{document}
