<?xml version="1.0"?>
<div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1"/>
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_1"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_1"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b3">s</font></span>)</span> where <font color="Olive" title="b2">r</font>, <font color="Olive" title="b3">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:32_1_1_1_1_1"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span> 
 ;<br/>

<a NAME="E3:32_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E2:32_1_1_1_1_1"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:32_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:32_1_1_1_1"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:32_1_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span>  is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E18"><i><font color="Green" title="E12">Lm9</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c5">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span> ;<br/><a NAME="E2:32_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_1_1_1_2_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:32_1_1_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 
;<br/>
<b>hence </b><a NAME="E3:32_1_1_1_2_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c6">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  ;<br/><a NAME="E4:32_1_1_1_2"/>
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> holds <br/>( ( for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b2">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> ) &amp;  ex <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b2">s</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_1_1_2_2"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">r0</font>, <font color="Maroon" title="c9">s0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E3:32_1_1_1_2_2"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">s0</font>
 <b>and </b><br/><a NAME="E4:32_1_1_1_2_2"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c8">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E5:32_1_1_1_2_2"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c9">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ;<br/>
<b>thus </b><a NAME="E6:32_1_1_1_2_2"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b1">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c7">r</font> holds <br/><font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_1_1_2_2_1"/>
<font color="Maroon" title="c10">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c7">r</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">t0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E3:32_1_1_1_2_2_1"/><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c11">t0</font>
 <b>and </b><br/><a NAME="E4:32_1_1_1_2_2_1"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c11">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">s0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:32_1_1_1_2_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="arytm_3.html#T87" title="ARYTM_3:th.87">ARYTM_3:87</a></i>;<br/>
<a NAME="E5:32_1_1_1_2_2_1"/>
<font color="Maroon" title="c11">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:32_1_1_1_2_2"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="arytm_2.html#E4:32_1_1_1_2_2_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E21">Lm17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:32_1_1_1_2_2_1"/>
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:32_1_1_1_2_2"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="arytm_2.html#E3:32_1_1_1_2_2_1"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>consider </b><font color="Maroon" title="c10">t0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E8:32_1_1_1_2_2"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E9:32_1_1_1_2_2"/><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c9">s0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">t0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:32_1_1_1_2_2"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="arytm_2.html#E17"><i><font color="Green" title="E11">Lm8</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1_2_2_2"/>
( <font color="Maroon" title="c8">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or <font color="Maroon" title="c8">r0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_2_2_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_2_2_2_1"/><i><font color="Green" title="E37">A11</font></i>: 
<font color="Maroon" title="c8">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c11">r1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E3:32_1_1_1_2_2_2_1"/><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c11">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E4:32_1_1_1_2_2_2_1"/><i><font color="Green" title="E39">A13</font></i>: 
<font color="Maroon" title="c11">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:32_1_1_1_2"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green" title="E20">Lm16</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font>
;<br/><b>thus </b><a NAME="E5:32_1_1_1_2_2_2_1"/>
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:32_1_1_1_2_2"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="arytm_2.html#E3:32_1_1_1_2_2_2_1"><i><font color="Green" title="E38">A12</font></i></a></i>;<br/><a NAME="E6:32_1_1_1_2_2_2_1"/><i><font color="Green" title="E40">A14</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:32_1_1_1_2_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="arytm_2.html#E1:32_1_1_1_2_2_2_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>;<br/><a NAME="E7:32_1_1_1_2_2_2_1"/><b>then </b><i><font color="Green" title="E41">A15</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></i>;<br/><a NAME="E8:32_1_1_1_2_2_2_1"/>
<font color="Maroon" title="c10">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E9:32_1_1_1_2_2"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></i>;<br/><a NAME="E9:32_1_1_1_2_2_2_1"/><b>then </b>
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:32_1_1_1_2_2_2_1"><i><font color="Green" title="E39">A13</font></i></a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a></i>;<br/><b>hence </b><a NAME="E10:32_1_1_1_2_2_2_1"/>
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:32_1_1_1_2_2_2_1"><i><font color="Green" title="E40">A14</font></i></a>, <a class="txt" href="arytm_2.html#E7:32_1_1_1_2_2_2_1"><i><font color="Green" title="E41">A15</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_2_2_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2_2_2_2"/><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c8">r0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font>
;<br/><b>thus </b><a NAME="E2:32_1_1_1_2_2_2_2"/>
<font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:32_1_1_1_2_2"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="arytm_2.html#E8:32_1_1_1_2_2"><i><font color="Green" title="E35">A9</font></i></a></i>;<br/><a NAME="E3:32_1_1_1_2_2_2_2"/>
<font color="Maroon" title="c9">s0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">t0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E9:32_1_1_1_2_2"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/><a NAME="E4:32_1_1_1_2_2_2_2"/><b>then </b><i><font color="Green" title="E38">A17</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:32_1_1_1_2_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="arytm_2.html#E1:32_1_1_1_2_2_2_2"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></i>;<br/><a NAME="E5:32_1_1_1_2_2_2_2"/>
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:32_1_1_1_2_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="arytm_2.html#E9:32_1_1_1_2_2"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/><b>hence </b><a NAME="E6:32_1_1_1_2_2_2_2"/>
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">t0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:32_1_1_1_2_2_2_2"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:32_1_1_1_2"/><b>then </b><i><font color="Green" title="E32">A18</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/>( ( for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) &amp;  ex <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b3">s</font> ) ) </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon" title="c7">r0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E7:32_1_1_1_2"/><i><font color="Green" title="E33">A19</font></i>: 
not <font color="Maroon" title="c7">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E36"><i><font color="Green" title="E30">Lm25</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">s0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E9:32_1_1_1_2"/><i><font color="Green" title="E34">A20</font></i>: 
not <font color="Maroon" title="c8">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E36"><i><font color="Green" title="E30">Lm25</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1_2_3"/>
<font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c8">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c10">s1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E3:32_1_1_1_2_3"/><i><font color="Green" title="E35">A21</font></i>: 
<font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c8">s0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">s1</font>
 <b>and </b><br/><a NAME="E4:32_1_1_1_2_3"/><i><font color="Green" title="E36">A22</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E5:32_1_1_1_2_3"/><i><font color="Green" title="E37">A23</font></i>: 
<font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ;<br/><a NAME="E6:32_1_1_1_2_3"/>
( <font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">r1</font> or <font color="Maroon" title="c8">s0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">s1</font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:32_1_1_1_2_3"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="arytm_3.html#T91" title="ARYTM_3:th.91">ARYTM_3:91</a></i>;<br/><b>hence </b><a NAME="E7:32_1_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E7:32_1_1_1_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="arytm_2.html#E9:32_1_1_1_2"><i><font color="Green" title="E34">A20</font></i></a>, <a class="txt" href="arytm_2.html#E4:32_1_1_1_2_3"><i><font color="Green" title="E36">A22</font></i></a>, <a class="txt" href="arytm_2.html#E5:32_1_1_1_2_3"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E21">Lm17</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E11:32_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 ;<br/><b>hence </b><a NAME="E12:32_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span>  is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:32_1_1_1_2"><i><font color="Green" title="E32">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
