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\begin{document}
\title{thm\_2Ehrat\_2ETRAT\_\_ADD\_\_SYM (TMSfU9ZrZcnojkJjajppEWq32AGyBR9HVG2)}
\maketitle
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ ty\_2Epair\_2Eprod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_2Ehrat\_2Etrat\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ehrat\_2Etrat\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeigchbgehpfcdfeehchbgehpfpffgbh}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
Let $ c\_2Ehrat\_2Etrat\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ehrat\_2Etrat\_\_add \in (((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeigchbgehpfcdfeehchbgehpfpfbgegeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in (ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum) .  \\
({\unicdcdadad} V1q \in (ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum) .  \\
((V0p=V1q){\unicdbdjdcd}(p~(ap~(ap~c\_2Ehrat\_2Etrat\_\_eq~V0p)~V1q)))))\end{array}
\label{hyp:ehigngpfcdfeigchbgehpfcdfeefcfbeefpfpffebfpfpfbeaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0h \in (ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum) .  \\
({\unicdcdadad} V1i \in (ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum) .  \\
((ap~(ap~c\_2Ehrat\_2Etrat\_\_add~V0h)~V1i)=(ap~(ap~c\_2Ehrat\_2Etrat\_\_add~\\
V1i)~V0h))))\end{array}
\label{hyp:ehigngpfcdfeigchbgehpfcdfeefcfbeefpfpfbeeeeepfpfdfjfnepfpffebf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeigchbgehpfcdfeefcfbeefpfpfbeeeeepfpfdfjfne}
$$\begin{array}{c}({\unicdcdadad} V0h \in (ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum) .  \\
({\unicdcdadad} V1i \in (ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum) .  \\
(p~(ap~(ap~c\_2Ehrat\_2Etrat\_\_eq~(ap~(ap~c\_2Ehrat\_2Etrat\_\_add~V0h)~\\
V1i))~(ap~(ap~c\_2Ehrat\_2Etrat\_\_add~V1i)~V0h)))))\end{array}$$
\end{theorem}
\end{document}
