\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{thm\_2Erat\_2ERAT\_\_EQUIV\_\_NMR\_\_LTZ\_\_IFF (TMQcY7Y8eC31mbRgrmrcAjd2cYMbVZ8B39k)}
\maketitle
\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}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
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 $ ty\_2Einteger\_2Eint : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Einteger\_2Eint}}\label{hyp:ogfgpfehjhpfcdfejgogehfghgfgchpfcdfejgogeh}
\end{equation}
Let $ c\_2Einteger\_2Eint\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeafpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeaf}
We define $\mname{c\_2Einteger\_2Eint\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_lt : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_lt \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfmgeh}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfmgeh}
We define $\mname{c\_2Einteger\_2Eint\_\_lt}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_lt}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfhgeh}
We define $\mname{c\_2Einteger\_2Eint\_\_gt}$ to be ${{\uniddcgcg} V0x \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1y \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_lt}}}~{V1y}})}}~{V0x}})}}}$.
\end{definition}
\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}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfmgfg}
We define $\mname{c\_2Einteger\_2Eint\_\_le}$ to be ${{\uniddcgcg} V0x \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1y \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_lt}}}~{V1y}})}}~{V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\end{definition}
Let $ c\_2Einteger\_2Eint\_\_of\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Einteger\_2Eint\_\_of\_\_num} \in {({{ty\_2Einteger\_2Eint}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfpgggpfpfogfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
Let $ ty\_2Efrac\_2Efrac : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Efrac\_2Efrac}}\label{hyp:ogfgpfehjhpfcdfeggchbgdgpfcdfeggchbgdg}
\end{equation}
Let $ c\_2Efrac\_2Erep\_\_frac : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Efrac\_2Erep\_\_frac \in ((ty\_2Epair\_2Eprod~ty\_2Einteger\_2Eint~\\
ty\_2Einteger\_2Eint)^{{ty\_2Efrac\_2Efrac}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggchbgdgpfcdfechfgahpfpfggchbgdg}
\end{equation}
Let $ c\_2Epair\_2ESND : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epair\_2ESND~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfedfoeee}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfegogng}
We define $\mname{c\_2Efrac\_2Efrac\_\_dnm}$ to be ${{\uniddcgcg} V0f \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{c\_2Epair\_2ESND}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{c\_2Efrac\_2Erep\_\_frac}}~{V0f}})}})}}$.
\end{definition}
Let $ c\_2Epair\_2EFST : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epair\_2EFST~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfegedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfogngch}
We define $\mname{c\_2Efrac\_2Efrac\_\_nmr}$ to be ${{\uniddcgcg} V0f \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{c\_2Epair\_2EFST}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{c\_2Efrac\_2Erep\_\_frac}}~{V0f}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_mul : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_mul \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:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfngfhmg}
\end{equation}
Let $ c\_2Einteger\_2Etint\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_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:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpffgbh}
\end{equation}
Let $ c\_2Einteger\_2Eint\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_ABS\_\_CLASS \in (ty\_2Einteger\_2Eint^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedf}
We define $\mname{c\_2Einteger\_2Eint\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})} .  {({{{ap}~{c\_2Einteger\_2Eint\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Einteger\_2Etint\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfngfhmg}
We define $\mname{c\_2Einteger\_2Eint\_\_mul}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_mul}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpffgbhfhjggh}
We define $\mname{c\_2Erat\_2Erat\_\_equiv}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {{\uniddcgcg} V1f2 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V1f2}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V1f2}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjdcd}(p~V1t2)){\unicdbdjdcd}(((p~\\
V1t2){\unicdbdjdcd}(p~V0t1)){\unicdbdjdcd}((p~V0t1){\unicdbdjded}(p~V1t2))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfbeoeefjedfjfnepfpfbeif}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{False}{\unicdbdjdcd}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegebemedfjeefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in ty\_2Efrac\_2Efrac .  (p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~\\
(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~(ap~\mname{c\_2Efrac\_2Efrac\_\_dnm}~\\
V0f))))\end{array}
\label{hyp:ehigngpfcdfeggchbgdgpfcdfegecfbedepfpfeeoeneafpedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
((\preoh (p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~V0x)~V1y))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_le}~\\
V1y)~V0x)))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfoepeefpfpfmeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
((p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~V0x)~V1y)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_le}~\\
V0x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfmeefpfpfjeneafpfpfmefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1q \in ty\_2Einteger\_2Eint .  \\
(((p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V0p)~V1q))){\unicdbdjded}(((\\
p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V0p)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V1q))){\unicdcdcdid}((p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~V0p)~(\\
ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~\\
V1q)~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0})))))){\unicdcdcdhd}((p~(\\
ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V0p)~\\
V1q))~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0}))){\unicdbdjded}(((p~(ap~\\
(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~\\
V0p)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~V1q)~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))){\unicdcdcdid}((p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~V0p)~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V1q))))))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfneffmepfpfdfjeheoepfpfdebedffedf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechbgehpfcdfecfbeefpfpffebfffjegfpfpfoenecfpfpfmeefkfpfpfjegege}
$$\begin{array}{c}({\unicdcdadad} V0a \in ty\_2Efrac\_2Efrac .  ({\unicdcdadad} V1b \in ty\_2Efrac\_2Efrac .  \\
((p~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_equiv}~V0a)~V1b)){\unicdbdjdcd}((p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~\\
(ap~\mname{c\_2Efrac\_2Efrac\_\_nmr}~V0a))~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~(ap~\mname{c\_2Efrac\_2Efrac\_\_nmr}~\\
V1b))~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0})))))))\end{array}$$
\end{theorem}
\end{document}
