\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{l20\_scmringi (TMb66c7t5ra1F9L1rTi4iZk5mLHja9XnDqD)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v13\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l6\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_scmringi : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_compos\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_7 : {\iota}$ be given.
Let $ k4\_finseq\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_finseq\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ l1\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_tarski : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ k4\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k4\_ordinal1 : {\iota}$ be given.
Let $ k7\_card\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_8 : {\iota}$ be given.
Let $ k2\_scm\_inst : {\iota}$ be given.
Let $ k2\_finseq\_4 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k2\_enumset1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ np\_\_3 : {\iota}$ be given.
Let $ np\_\_4 : {\iota}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ np\_\_6 : {\iota}$ be given.
Let $ k12\_finseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_5 : {\iota}$ be given.
Let $ k7\_scmringi : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ v1\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_finseq\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k9\_finseq\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k10\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k5\_finseq\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_compos\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l5\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~(k1\_scmringi~X0)){\unicdbdjdcd}(\preoh (\preoh (X1 \in k1\_tarski~(k3\_xtuple\_0~\\
k6\_numbers~k1\_xboole\_0~k1\_xboole\_0)){\unicdcdcdhd}(k4\_xtuple\_0~X1=k6\_numbers)){\unicdcdcdhd}\\
((\preoh (X1 \in ReplSep3~(toset~({\uniddcgcg} X2 : \iota .  m2\_subset\_1~X2~k4\_ordinal1~\\
(k7\_card\_1~np\_\_8)))~({\uniddcgcg} X2 : \iota .  toset~({\uniddcgcg} X3 : \iota .  m1\_subset\_1~\\
X3~k2\_scm\_inst))~({\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  toset~({\uniddcgcg} X4 : \iota .  \\
m1\_subset\_1~X4~k2\_scm\_inst))~({\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  {\uniddcgcg} X4 : \\
\iota .  X2 \in k2\_enumset1~np\_\_1~np\_\_2~np\_\_3~np\_\_4)~({\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \\
\iota .  {\uniddcgcg} X4 : \iota .  k3\_xtuple\_0~X2~k1\_xboole\_0~(k2\_finseq\_4~k2\_scm\_inst~\\
X3~X4))){\unicdcdcdhd}(\preoh (k4\_xtuple\_0~X1{\unicdcdgdad}np\_\_1){\unicdcdcdhd}((k4\_xtuple\_0~X1{\unicdcdgdad}np\_\_2){\unicdcdcdhd}\\
((k4\_xtuple\_0~X1{\unicdcdgdad}np\_\_3){\unicdcdcdhd}(k4\_xtuple\_0~X1{\unicdcdgdad}np\_\_4))))){\unicdcdcdhd}((\preoh (X1 \in \\
ReplSep~(toset~({\uniddcgcg} X2 : \iota .  m1\_subset\_1~X2~k5\_numbers))~({\uniddcgcg} X2 : \\
\iota .  True)~({\uniddcgcg} X2 : \iota .  k3\_xtuple\_0~np\_\_6~(k12\_finseq\_1~k5\_numbers~\\
X2)~k1\_xboole\_0)){\unicdcdcdhd}(k4\_xtuple\_0~X1=np\_\_6)){\unicdcdcdhd}((\preoh (X1 \in ReplSep2~\\
(toset~({\uniddcgcg} X2 : \iota .  m1\_subset\_1~X2~k5\_numbers))~({\uniddcgcg} X2 : \iota .  \\
toset~({\uniddcgcg} X3 : \iota .  m1\_subset\_1~X3~k2\_scm\_inst))~({\uniddcgcg} X2 : \iota .  \\
{\uniddcgcg} X3 : \iota .  True)~({\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  k3\_xtuple\_0~np\_\_7~\\
(k12\_finseq\_1~k5\_numbers~X2)~(k12\_finseq\_1~k2\_scm\_inst~X3))){\unicdcdcdhd}\\
(k4\_xtuple\_0~X1=np\_\_7)){\unicdcdcdhd}(\preoh (X1 \in ReplSep2~(toset~({\uniddcgcg} X2 : \iota .  \\
m1\_subset\_1~X2~k2\_scm\_inst))~({\uniddcgcg} X2 : \iota .  toset~({\uniddcgcg} X3 : \iota .  \\
m1\_subset\_1~X3~(u1\_struct\_0~X0)))~({\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  True)~\\
({\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  k3\_xtuple\_0~np\_\_5~k1\_xboole\_0~(k7\_scmringi~\\
X0~X2~X3))){\unicdcdcdhd}(k4\_xtuple\_0~X1=np\_\_5)))))))\end{array}
\label{hyp:ehhdpfdhdgngchjgoghgjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}((v1\_funct\_1~X1){\unicdcdcdhd}(v1\_finseq\_1~\\
X1))){\unicdbdjdcd}((X1=k9\_finseq\_1~X0){\unicdbdjded}((k4\_finseq\_1~X1=k2\_finseq\_1~np\_\_1){\unicdcdcdhd}\\
(k10\_xtuple\_0~X1=k1\_tarski~X0)))\end{array}
\label{hyp:ehddidpfggjgogdhfgbhpfbd}
\end{equation}
Assume the following.
\begin{equation}
{\preoh{{{v1\_xboole\_0}~{np\_\_7}}}}\label{hyp:dhahdghdpfcgpgpgmgfg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{k9\_finseq\_1}~{X0}}={{k5\_finseq\_1}~{X0}}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgjdpfggjgogdhfgbhpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k6\_numbers}={k1\_xboole\_0}}\label{hyp:chfgegfgggjgogjgehjgpgogpflggdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{k5\_numbers}={k4\_ordinal1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_compos\_0~X0)){\unicdcdcdhd}\\
(m1\_subset\_1~X1~X0)){\unicdbdjdcd}(k2\_compos\_0~X0~X1=k4\_xtuple\_0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfdgpgngahpgdhpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(m1\_subset\_1~X1~X0)){\unicdbdjdcd}\\
(k12\_finseq\_1~X0~X1=k5\_finseq\_1~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdcdpfggjgogdhfgbhpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  k5\_xtuple\_0~(k3\_xtuple\_0~X0~\\
X1~X2)=X1\end{array}
\label{hyp:chegfdpfihehfhahmgfgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{v1\_finseq\_1}~{({{k5\_finseq\_1}~{X0}})}}}\label{hyp:ggdghdpfggjgogdhfgbhpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{({\preoh{{{v1\_xboole\_0}~{k4\_ordinal1}}}})}{\unicdcdcdhd}{({{v3\_ordinal1}~{k4\_ordinal1}})}}\label{hyp:ggdggdpfpgchegjgogbgmgbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v2\_rlvect\_1~\\
X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0))))))))){\unicdbdjdcd}\\
((\preoh v1\_xboole\_0~(k1\_scmringi~X0)){\unicdcdcdhd}(v1\_compos\_0~(k1\_scmringi~\\
X0)))\end{array}
\label{hyp:ggdgedpfdhdgngchjgoghgjg}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_xboole\_0}~{k1\_xboole\_0}}\label{hyp:ggdgbdpfihcgpgpgmgfgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l6\_algstr\_0}~{X0}})}{\unicdbdjdcd}{({{({{l2\_algstr\_0}~{X0}})}{\unicdcdcdhd}{({{l5\_algstr\_0}~{X0}})}})}}}\label{hyp:egehpfmggdpfbgmghgdhehchpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l2\_struct\_0}~{X0}})}{\unicdbdjdcd}{({{l1\_struct\_0}~{X0}})}}}\label{hyp:egehpfmgcdpfdhehchfhdgehpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l2\_algstr\_0}~{X0}})}{\unicdbdjdcd}{({{({{l2\_struct\_0}~{X0}})}{\unicdcdcdhd}{({{l1\_algstr\_0}~{X0}})}})}}}\label{hyp:egehpfmgcdpfbgmghgdhehchpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_relat\_1~(k9\_finseq\_1~X0)){\unicdcdcdhd}(v1\_funct\_1~(k9\_finseq\_1~\\
X0))\end{array}
\label{hyp:egehpflgjdpfggjgogdhfgbhpfbd}
\end{equation}
\begin{theorem}\label{thm:mgcdadpfdhdgngchjgoghgjg}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v2\_rlvect\_1~\\
X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0))))))))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k1\_scmringi~X0)){\unicdbdjdcd}((k2\_compos\_0~\\
(k1\_scmringi~X0)~X1=np\_\_7){\unicdbdjdcd}(k4\_finseq\_1~(k5\_xtuple\_0~X1)=k2\_finseq\_1~\\
np\_\_1)))\end{array}$$
\end{theorem}
\end{document}
