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

<b>let </b><font color="Maroon" title="c4">A</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E58">A1</font></i>: 
<font color="Maroon" title="c4">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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1"/>
( <font color="Maroon" title="c4">A</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">s</font> where <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="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> </span>}</span>  where <font color="Olive" title="b1">t</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">t</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>}</span>  or  not <font color="Maroon" title="c4">A</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">s</font> where <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="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> </span>}</span>  where <font color="Olive" title="b1">t</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">t</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>}</span>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1"><b>suppose </b></a><a NAME="E1:60_1_1"/>
<font color="Maroon" title="c4">A</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">s</font> where <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="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> </span>}</span>  where <font color="Olive" title="b1">t</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">t</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>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">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="E3:60_1_1"/><i><font color="Green" title="E59">A2</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">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">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> </span>}</span> 
 <b>and </b><br/><a NAME="E4:60_1_1"/><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c5">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/><b>consider </b><font color="Maroon" title="c6">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="E6:60_1_1"/><i><font color="Green" title="E61">A4</font></i>: 
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">s0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:60_1_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="arytm_3.html#T60" title="ARYTM_3:th.60">ARYTM_3:60</a></i>;<br/><b>set </b><font color="Maroon" title="c7">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">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">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font> </span>}</span> ;<br/><a NAME="E7:60_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">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">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</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="60_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:60_1_1_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">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">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font> </span>}</span> 
 ;<br/>

<a NAME="E2:60_1_1_1"/><b>then </b>
 ex <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 <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font> )
 
;<br/>
<b>hence </b><a NAME="E3:60_1_1_1"/>
<font color="Maroon" title="c8">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="c8">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">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">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</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="E9:60_1_1"/>
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>   holds <br/>(  not <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s0</font> or  not <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font> )
 ;<br/><a NAME="E10:60_1_1"/><b>then </b>
not <font color="Maroon" title="c6">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/><a NAME="E11:60_1_1"/><b>then </b><i><font color="Green" title="E62">A5</font></i>: 
<font color="Maroon" title="c8">B</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/><a NAME="E12:60_1_1"/>
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="c8">B</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="c8">B</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="c8">B</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="60_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="E1:60_1_1_2"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/>

<a NAME="E2:60_1_1_2"/><b>then </b><i><font color="Green" title="E63">A6</font></i>: 
 ex <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 <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font> )
 
;<br/>
<b>thus </b><a NAME="E3:60_1_1_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="c9">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="c8">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_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="E1:60_1_1_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="c9">r</font>
 ;<br/>

<a NAME="E2:60_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E2:60_1_1_2"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></i>;<br/>
<b>hence </b><a NAME="E3:60_1_1_2_1"/>
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/>


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

<b>consider </b><font color="Maroon" title="c10">t</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="E5:60_1_1_2"/><i><font color="Green" title="E64">A7</font></i>: 
<font color="Maroon" title="c9">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">t</font>
 <b>and </b><br/><a NAME="E6:60_1_1_2"/><i><font color="Green" title="E65">A8</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:60_1_1_2"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="arytm_3.html#T101" title="ARYTM_3:th.101">ARYTM_3:101</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">t</font>
;<br/>

<b>thus </b><a NAME="E7:60_1_1_2"/>
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:60_1_1_2"><i><font color="Green" title="E65">A8</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:60_1_1_2"/>
<font color="Maroon" title="c9">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">t</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:60_1_1_2"><i><font color="Green" title="E64">A7</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:60_1_1"/><b>then </b>
<font color="Maroon" title="c8">B</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>then reconsider </b><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c8">B</font> as    <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#E11:60_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E15:60_1_1"/><i><font color="Green" title="E63">A9</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">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">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:60_1_1_3"/>
<font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">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">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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="E1:60_1_1_3_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</font>
 ;<br/>

<a NAME="E2:60_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c10">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="c4">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="c9">B</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c11">r1</font>, <font color="Maroon" title="c12">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="E4:60_1_1_3_1"/><i><font color="Green" title="E64">A10</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s1</font>
 <b>and </b><br/><a NAME="E5:60_1_1_3_1"/><i><font color="Green" title="E65">A11</font></i>: 
<font color="Maroon" title="c11">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>and </b><br/><a NAME="E6:60_1_1_3_1"/><i><font color="Green" title="E66">A12</font></i>: 
<font color="Maroon" title="c12">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/>
<a NAME="E7:60_1_1_3_1"/><i><font color="Green" title="E67">A13</font></i>: 
 ex <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 <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E3:60_1_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="arytm_2.html#E5:60_1_1_3_1"><i><font color="Green" title="E65">A11</font></i></a></i>;<br/>
<a NAME="E8:60_1_1_3_1"/><i><font color="Green" title="E68">A14</font></i>: 
 ex <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 <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s1</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E6:60_1_1_3_1"><i><font color="Green" title="E66">A12</font></i></a></i>;<br/>
<a NAME="E9:60_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c12">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">s0</font>
 
;<br/>
<a NAME="E10:60_1_1_3_1"/><b>then </b><i><font color="Green" title="E69">A15</font></i>: 
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E4:60_1_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></i>;<br/>
<a NAME="E11:60_1_1_3_1"/>
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s1</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E8:60_1_1_3_1"><i><font color="Green" title="E68">A14</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/>
<a NAME="E12:60_1_1_3_1"/><b>then </b><i><font color="Green" title="E70">A16</font></i>: 
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E10:60_1_1_3_1"><i><font color="Green" title="E69">A15</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/>
<a NAME="E13:60_1_1_3_1"/>
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s1</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s1</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E7:60_1_1_3_1"><i><font color="Green" title="E67">A13</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/>
<a NAME="E14:60_1_1_3_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="c12">s1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E12:60_1_1_3_1"><i><font color="Green" title="E70">A16</font></i></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></i>;<br/>
<b>hence </b><a NAME="E15:60_1_1_3_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">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">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:60_1_1_3_1"><i><font color="Green" title="E64">A10</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c10">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="E2:60_1_1_3"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">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">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">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="E4:60_1_1_3"/><i><font color="Green" title="E64">A17</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s1</font>
 <b>and </b><br/><a NAME="E5:60_1_1_3"/><i><font color="Green" title="E65">A18</font></i>: 
<font color="Maroon" title="c11">s1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">s0</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c12">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="E7:60_1_1_3"/><i><font color="Green" title="E66">A19</font></i>: 
<font color="Maroon" title="c11">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">t0</font>
 <b>and </b><br/><a NAME="E8:60_1_1_3"/><i><font color="Green" title="E67">A20</font></i>: 
<font color="Maroon" title="c12">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">s0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:60_1_1_3"><i><font color="Green" title="E65">A18</font></i></a>, <a class="ref" href="arytm_3.html#T87" title="ARYTM_3:th.87">ARYTM_3:87</a></i>;<br/>
<a NAME="E9:60_1_1_3"/>
<font color="Maroon" title="c12">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:60_1_1_3"><i><font color="Green" title="E65">A18</font></i></a>, <a class="txt" href="arytm_2.html#E7:60_1_1_3"><i><font color="Green" title="E66">A19</font></i></a></i>;<br/>
<a NAME="E10:60_1_1_3"/><b>then </b>
<font color="Maroon" title="c12">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E8:60_1_1_3"><i><font color="Green" title="E67">A20</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/>
<a NAME="E11:60_1_1_3"/><b>then </b>
<font color="Maroon" title="c12">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c13">t1</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="E13:60_1_1_3"/><i><font color="Green" title="E68">A21</font></i>: 
<font color="Maroon" title="c13">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>and </b><br/><a NAME="E14:60_1_1_3"/><i><font color="Green" title="E69">A22</font></i>: 
<font color="Maroon" title="c12">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c13">t1</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E17"><i><font color="Green" title="E12">Lm8</font></i></a></i>;<br/>
<a NAME="E15:60_1_1_3"/>
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c13">t1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c5">r0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E14:60_1_1_3"><i><font color="Green" title="E69">A22</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">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="E17:60_1_1_3"/><i><font color="Green" title="E70">A23</font></i>: 
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">r1</font>
 <b>and </b><br/><a NAME="E18:60_1_1_3"/><i><font color="Green" title="E71">A24</font></i>: 
<font color="Maroon" title="c14">r1</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T87" title="ARYTM_3:th.87">ARYTM_3:87</a></i>;<br/>
<a NAME="E19:60_1_1_3"/>
<font color="Maroon" title="c12">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">t1</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E14:60_1_1_3"><i><font color="Green" title="E69">A22</font></i></a></i>;<br/>
<a NAME="E20:60_1_1_3"/><b>then </b>
<font color="Maroon" title="c14">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E4:60_1_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="arytm_2.html#E17:60_1_1_3"><i><font color="Green" title="E70">A23</font></i></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></i>;<br/>
<a NAME="E21:60_1_1_3"/><b>then </b>
<font color="Maroon" title="c14">r1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E18:60_1_1_3"><i><font color="Green" title="E71">A24</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/>
<a NAME="E22:60_1_1_3"/><b>then </b>
<font color="Maroon" title="c14">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E3:60_1_1"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/>
<a NAME="E23:60_1_1_3"/><b>then </b>
<font color="Maroon" title="c10">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="c4">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="c9">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E4:60_1_1_3"><i><font color="Green" title="E64">A17</font></i></a>, <a class="txt" href="arytm_2.html#E7:60_1_1_3"><i><font color="Green" title="E66">A19</font></i></a>, <a class="txt" href="arytm_2.html#E13:60_1_1_3"><i><font color="Green" title="E68">A21</font></i></a>, <a class="txt" href="arytm_2.html#E17:60_1_1_3"><i><font color="Green" title="E70">A23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E24:60_1_1_3"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E16:60_1_1"/>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Olive" title="b1">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">rone</font> &amp;  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">s</font> where <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="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t0</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E17:60_1_1"/>
 ex <font color="Olive" title="b1">B</font> being   <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>  st <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:60_1_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="arytm_2.html#E15:60_1_1"><i><font color="Green" title="E63">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2"><b>suppose </b></a><a NAME="E1:60_1_2"/><i><font color="Green" title="E59">A25</font></i>: 
not <font color="Maroon" title="c4">A</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">s</font> where <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="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> </span>}</span>  where <font color="Olive" title="b1">t</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">t</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>}</span> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c5">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y'</font> where <font color="Olive" title="b1">y'</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>  :  ex <font color="Olive" title="b2">x'</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/>( not <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">y'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  ) </span>}</span> ;<br/><a NAME="E2:60_1_2"/><i><font color="Green" title="E60">A26</font></i>: 
<font color="Maroon" title="c4">A</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> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">x0</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="E4:60_1_2"/><i><font color="Green" title="E61">A27</font></i>: 
not <font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T49" title="SUBSET_1:th.49">SUBSET_1:49</a></i>;<br/><a NAME="E5:60_1_2"/>
<font color="Maroon" title="c6">x0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <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="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>;<br/><a NAME="E6:60_1_2"/><b>then </b>
( <font color="Maroon" title="c6">x0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  &amp; <font color="Maroon" title="c6">x0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  )
 <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="E7:60_1_2"/><b>then </b><i><font color="Green" title="E62">A28</font></i>: 
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y'</font> where <font color="Olive" title="b1">y'</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>  :  ex <font color="Olive" title="b2">x'</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/>( not <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">y'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:60_1_2"><i><font color="Green" title="E61">A27</font></i></a></i>;<br/><a NAME="E8:60_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y'</font> where <font color="Olive" title="b1">y'</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>  :  ex <font color="Olive" title="b2">x'</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/>( not <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">y'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  ) </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="60_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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="E1:60_1_2_1"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y'</font> where <font color="Olive" title="b1">y'</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>  :  ex <font color="Olive" title="b2">x'</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/>( not <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">y'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  ) </span>}</span> 
 ;<br/>

<a NAME="E2:60_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">y'</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">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e</font> &amp;  ex <font color="Olive" title="b2">x'</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/>( not <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">y'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  ) )
 
;<br/>
<b>hence </b><a NAME="E3:60_1_2_1"/>
<font color="Maroon" title="c7">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="c7">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y'</font> where <font color="Olive" title="b1">y'</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>  :  ex <font color="Olive" title="b2">x'</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/>( not <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">y'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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>  <b>by </b><i><a class="txt" href="arytm_2.html#E7:60_1_2"><i><font color="Green" title="E62">A28</font></i></a></i>;<br/><a NAME="E10:60_1_2"/><i><font color="Green" title="E63">A29</font></i>: 
<font color="Maroon" title="c4">A</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> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">x'</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="E12:60_1_2"/><i><font color="Green" title="E64">A30</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:60"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E13:60_1_2"/>
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">x'</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="E14:60_1_2"/><b>then </b><i><font color="Green" title="E65">A31</font></i>: 
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:60_1_2"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="arytm_2.html#E12:60_1_2"><i><font color="Green" title="E64">A30</font></i></a>, <a class="txt" href="arytm_2.html#E49"><i><font color="Green" title="E40">Lm30</font></i></a></i>;<br/><a NAME="E15:60_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="c7">B</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="c7">B</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="c7">B</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="60_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="E1:60_1_2_2"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ;<br/>

<a NAME="E2:60_1_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">y'</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="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y'</font> &amp;  ex <font color="Olive" title="b2">x'</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/>( not <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">y'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  ) )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">x'</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="E4:60_1_2_2"/><i><font color="Green" title="E66">A32</font></i>: 
not <font color="Maroon" title="c10">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>and </b><br/><a NAME="E5:60_1_2_2"/><i><font color="Green" title="E67">A33</font></i>: 
<font color="Maroon" title="c10">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 ;<br/>
<b>thus </b><a NAME="E6:60_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="c9">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="c7">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="E1:60_1_2_2_1"/>
<font color="Maroon" title="c11">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">r</font>
 ;<br/>

<a NAME="E2:60_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c10">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c11">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">r</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/>
<a NAME="E3:60_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c10">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c11">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:60_1_2_2"><i><font color="Green" title="E67">A33</font></i></a>, <a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></i>;<br/>
<b>hence </b><a NAME="E4:60_1_2_2_1"/>
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:60_1_2_2"><i><font color="Green" title="E66">A32</font></i></a></i>;<br/>


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

<a NAME="E7:60_1_2_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">s</font> where <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="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> </span>}</span>  where <font color="Olive" title="b1">t</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">t</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>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:60_1_2"><i><font color="Green" title="E59">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">x''</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="E9:60_1_2_2"/><i><font color="Green" title="E68">A34</font></i>: 
not <font color="Maroon" title="c11">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>and </b><br/><a NAME="E10:60_1_2_2"/><i><font color="Green" title="E69">A35</font></i>: 
<font color="Maroon" title="c11">x''</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">x'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E12:60_1_2"><i><font color="Green" title="E64">A30</font></i></a>, <a class="txt" href="arytm_2.html#E4:60_1_2_2"><i><font color="Green" title="E66">A32</font></i></a>, <a class="txt" href="arytm_2.html#E29"><i><font color="Green" title="E24">Lm19</font></i></a></i>;<br/>
<a NAME="E11:60_1_2_2"/><i><font color="Green" title="E70">A36</font></i>: 
( <font color="Maroon" title="c11">x''</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">x'</font> &amp; <font color="Maroon" title="c11">x''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">x'</font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E10:60_1_2_2"><i><font color="Green" title="E69">A35</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">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> <b> such that </b><br/><a NAME="E13:60_1_2_2"/><i><font color="Green" title="E71">A37</font></i>: 
 <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E14:60_1_2"><i><font color="Green" title="E65">A31</font></i></a>, <a class="txt" href="arytm_2.html#E9:60_1_2_2"><i><font color="Green" title="E68">A34</font></i></a>, <a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">s</font>
;<br/>

<a NAME="E14:60_1_2_2"/>
<font color="Maroon" title="c11">x''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E13:60_1_2_2"><i><font color="Green" title="E71">A37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:60_1_2_2"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E9:60_1_2_2"><i><font color="Green" title="E68">A34</font></i></a></i>;<br/>

<a NAME="E16:60_1_2_2"/>
<font color="Maroon" title="c11">x''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">r</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E10:60_1_2_2"><i><font color="Green" title="E69">A35</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/>
<a NAME="E17:60_1_2_2"/><b>then </b>
<font color="Maroon" title="c11">x''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c11">x''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:60_1_2_2"><i><font color="Green" title="E67">A33</font></i></a>, <a class="txt" href="arytm_2.html#E13:60_1_2_2"><i><font color="Green" title="E71">A37</font></i></a>, <a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></i>;<br/>
<a NAME="E18:60_1_2_2"/><b>then </b><i><font color="Green" title="E72">A38</font></i>: 
<font color="Maroon" title="c9">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c12">s</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E14:60_1_2"><i><font color="Green" title="E65">A31</font></i></a>, <a class="txt" href="arytm_2.html#E9:60_1_2_2"><i><font color="Green" title="E68">A34</font></i></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></i>;<br/>
<a NAME="E19:60_1_2_2"/><i><font color="Green" title="E73">A39</font></i>: 
<font color="Maroon" title="c11">x''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E10:60_1_2_2"><i><font color="Green" title="E69">A35</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/>
<a NAME="E20:60_1_2_2"/><i><font color="Green" title="E74">A40</font></i>: 
<font color="Maroon" title="c12">s</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#E13:60_1_2_2"><i><font color="Green" title="E71">A37</font></i></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_2_2_2"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font>
 ;<br/><a NAME="E2:60_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c11">x''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:60_1_2_2"><i><font color="Green" title="E67">A33</font></i></a>, <a class="txt" href="arytm_2.html#E13:60_1_2_2"><i><font color="Green" title="E71">A37</font></i></a>, <a class="txt" href="arytm_2.html#E19:60_1_2_2"><i><font color="Green" title="E73">A39</font></i></a>, <a class="ref" href="arytm_3.html#T73" title="ARYTM_3:th.73">ARYTM_3:73</a></i>;<br/><b>hence </b><a NAME="E3:60_1_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E11:60_1_2_2"><i><font color="Green" title="E70">A36</font></i></a>, <a class="txt" href="arytm_2.html#E20:60_1_2_2"><i><font color="Green" title="E74">A40</font></i></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E22:60_1_2_2"/>
<font color="Maroon" title="c9">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">s</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E18:60_1_2_2"><i><font color="Green" title="E72">A38</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><a NAME="E16:60_1_2"/><b>then </b><i><font color="Green" title="E66">A41</font></i>: 
<font color="Maroon" title="c7">B</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="c9">x'</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="E18:60_1_2"/><i><font color="Green" title="E67">A42</font></i>: 
<font color="Maroon" title="c9">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>and </b><br/><a NAME="E19:60_1_2"/><i><font color="Green" title="E68">A43</font></i>: 
<font color="Maroon" title="c9">x'</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#E1:60"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="arytm_2.html#E10:60_1_2"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="arytm_2.html#E70"><i><font color="Green" title="E56">Lm43</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">y'</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="E21:60_1_2"/><i><font color="Green" title="E69">A44</font></i>: 
<font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E19:60_1_2"><i><font color="Green" title="E68">A43</font></i></a>, <a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_2_3"/>
<font color="Maroon" title="c10">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ;<br/><a NAME="E2:60_1_2_3"/><b>then </b>
 ex <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 <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font> &amp;  ex <font color="Olive" title="b2">x'</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/>( not <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  ) )
 ;<br/><b>then consider </b><font color="Maroon" title="c11">z'</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="E4:60_1_2_3"/><i><font color="Green" title="E70">A45</font></i>: 
not <font color="Maroon" title="c11">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>and </b><br/><a NAME="E5:60_1_2_3"/><i><font color="Green" title="E71">A46</font></i>: 
<font color="Maroon" title="c11">z'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c10">y'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 ;<br/><a NAME="E6:60_1_2_3"/>
<font color="Maroon" title="c10">y'</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#E21:60_1_2"><i><font color="Green" title="E69">A44</font></i></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>;<br/><a NAME="E7:60_1_2_3"/><b>then </b>
<font color="Maroon" title="c11">z'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">x'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E21:60_1_2"><i><font color="Green" title="E69">A44</font></i></a>, <a class="txt" href="arytm_2.html#E5:60_1_2_3"><i><font color="Green" title="E71">A46</font></i></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></i>;<br/><b>hence </b><a NAME="E8:60_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:60_1_2"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="arytm_2.html#E18:60_1_2"><i><font color="Green" title="E67">A42</font></i></a>, <a class="txt" href="arytm_2.html#E4:60_1_2_3"><i><font color="Green" title="E70">A45</font></i></a>, <a class="txt" href="arytm_2.html#E49"><i><font color="Green" title="E40">Lm30</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E23:60_1_2"/><b>then </b>
<font color="Maroon" title="c7">B</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/><a NAME="E24:60_1_2"/><b>then </b>
not <font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">B</font> = <font color="Maroon" title="c7">B</font> as    <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#E16:60_1_2"><i><font color="Green" title="E66">A41</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">B</font>
;<br/><a NAME="E26:60_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>  holds <br/> ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c11">B</font> iff <font color="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:60_1_2_4_1"/>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c11">B</font>
 ;<br/><a NAME="E2:60_1_2_4_1"/><b>then </b>
<font color="Maroon" title="c12">r</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">s</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c13">s</font>, <font color="Maroon" title="c14">t</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="E4:60_1_2_4_1"/><i><font color="Green" title="E70">A47</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">t</font>
 <b>and </b><br/><a NAME="E5:60_1_2_4_1"/><i><font color="Green" title="E71">A48</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>and </b><br/><a NAME="E6:60_1_2_4_1"/><i><font color="Green" title="E72">A49</font></i>: 
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font>
 ;<br/><a NAME="E7:60_1_2_4_1"/>
 ex <font color="Olive" title="b1">z'</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">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">t</font> &amp;  ex <font color="Olive" title="b2">x'</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/>( not <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">z'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  ) )
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:60_1_2_4_1"><i><font color="Green" title="E72">A49</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">x'</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="E9:60_1_2_4_1"/><i><font color="Green" title="E73">A50</font></i>: 
not <font color="Maroon" title="c15">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>and </b><br/><a NAME="E10:60_1_2_4_1"/><i><font color="Green" title="E74">A51</font></i>: 
<font color="Maroon" title="c15">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 ;<br/><a NAME="E11:60_1_2_4_1"/>
<font color="Maroon" title="c13">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c15">x'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:60_1_2"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="arytm_2.html#E5:60_1_2_4_1"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="arytm_2.html#E9:60_1_2_4_1"><i><font color="Green" title="E73">A50</font></i></a>, <a class="txt" href="arytm_2.html#E49"><i><font color="Green" title="E40">Lm30</font></i></a></i>;<br/><a NAME="E12:60_1_2_4_1"/><b>then </b><i><font color="Green" title="E75">A52</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c15">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">t</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/><a NAME="E13:60_1_2_4_1"/><b>then </b><i><font color="Green" title="E76">A53</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:60_1_2_4_1"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="arytm_2.html#E10:60_1_2_4_1"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_2_4_1_1"/><i><font color="Green" title="E77">A54</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 ;<br/><a NAME="E2:60_1_2_4_1_1"/><b>then </b><i><font color="Green" title="E78">A55</font></i>: 
<font color="Maroon" title="c14">t</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#E4:60_1_2_4_1"><i><font color="Green" title="E70">A47</font></i></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>;<br/><a NAME="E3:60_1_2_4_1_1"/>
<font color="Maroon" title="c13">s</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">t</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:60_1_2_4_1"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="arytm_2.html#E10:60_1_2_4_1"><i><font color="Green" title="E74">A51</font></i></a>, <a class="txt" href="arytm_2.html#E12:60_1_2_4_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="arytm_2.html#E1:60_1_2_4_1_1"><i><font color="Green" title="E77">A54</font></i></a>, <a class="ref" href="arytm_3.html#T73" title="ARYTM_3:th.73">ARYTM_3:73</a></i>;<br/><b>hence </b><a NAME="E4:60_1_2_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:60_1_2_4_1"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="arytm_2.html#E9:60_1_2_4_1"><i><font color="Green" title="E73">A50</font></i></a>, <a class="txt" href="arytm_2.html#E2:60_1_2_4_1_1"><i><font color="Green" title="E78">A55</font></i></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:60_1_2_4_1"/>
<font color="Maroon" title="c12">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E13:60_1_2_4_1"><i><font color="Green" title="E76">A53</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>

<b>assume </b><a NAME="E2:60_1_2_4"/><i><font color="Green" title="E70">A56</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 ;<br/>

<a NAME="E3:60_1_2_4"/><b>then </b><i><font color="Green" title="E71">A57</font></i>: 
( <font color="Maroon" title="c12">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  &amp; <font color="Maroon" title="c12">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1_2_4_2"/>
( <font color="Maroon" title="c12">r</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="c12">r</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="60_1_2_4_2_1"><b>suppose </b></a><a NAME="E1:60_1_2_4_2_1"/>
<font color="Maroon" title="c12">r</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="E2:60_1_2_4_2_1"/><b>then </b>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>;<br/><a NAME="E3:60_1_2_4_2_1"/><b>then </b>
<font color="Maroon" title="c12">r</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">s</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E7:60_1_2"><i><font color="Green" title="E62">A28</font></i></a>, <a class="txt" href="arytm_2.html#E14:60_1_2"><i><font color="Green" title="E65">A31</font></i></a></i>;<br/><b>hence </b><a NAME="E4:60_1_2_4_2_1"/>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c11">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_4_2_2"><b>suppose </b></a><a NAME="E1:60_1_2_4_2_2"/>
<font color="Maroon" title="c12">r</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>then consider </b><font color="Maroon" title="c13">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> <b> such that </b><br/><a NAME="E3:60_1_2_4_2_2"/><i><font color="Green" title="E72">A58</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c13">r'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">dr</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="E5:60_1_2_4_2_2"/><i><font color="Green" title="E73">A59</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c14">dr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:60_1_2_4"><i><font color="Green" title="E70">A56</font></i></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></i>;<br/><b>set </b><font color="Maroon" title="c15">rr</font> = <font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font>;<br/><b>consider </b><font color="Maroon" title="c16">xx</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="E7:60_1_2_4_2_2"/><i><font color="Green" title="E74">A60</font></i>: 
not <font color="Maroon" title="c16">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:60_1_2"><i><font color="Green" title="E60">A26</font></i></a>, <a class="ref" href="subset_1.html#T49" title="SUBSET_1:th.49">SUBSET_1:49</a></i>;<br/><a NAME="E8:60_1_2_4_2_2"/>
<font color="Maroon" title="c14">dr</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#E3:60_1_2_4"><i><font color="Green" title="E71">A57</font></i></a>, <a class="txt" href="arytm_2.html#E5:60_1_2_4_2_2"><i><font color="Green" title="E73">A59</font></i></a>, <a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a></i>;<br/><a NAME="E9:60_1_2_4_2_2"/><b>then </b>
<font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</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#E19:60_1_2"><i><font color="Green" title="E68">A43</font></i></a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">n0</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="E11:60_1_2_4_2_2"/><i><font color="Green" title="E75">A61</font></i>: 
<font color="Maroon" title="c17">n0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>and </b><br/><a NAME="E12:60_1_2_4_2_2"/><i><font color="Green" title="E76">A62</font></i>: 
<font color="Maroon" title="c16">xx</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c17">n0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T103" title="ARYTM_3:th.103">ARYTM_3:103</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>$1 <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>;<br/><a NAME="E13:60_1_2_4_2_2"/><i><font color="Green" title="E77">A63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> ]
 <b>by </b><i><a class="txt" href="arytm_2.html#E14:60_1_2"><i><font color="Green" title="E65">A31</font></i></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>;<br/><a NAME="E14:60_1_2_4_2_2"/><i><font color="Green" title="E78">A64</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">n0</font>]
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:60_1_2"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="arytm_2.html#E7:60_1_2_4_2_2"><i><font color="Green" title="E74">A60</font></i></a>, <a class="txt" href="arytm_2.html#E12:60_1_2_4_2_2"><i><font color="Green" title="E76">A62</font></i></a>, <a class="txt" href="arytm_2.html#E49"><i><font color="Green" title="E40">Lm30</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c18">n1</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="E16:60_1_2_4_2_2"/>
<font color="Maroon" title="c18">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>and </b><br/><a NAME="E17:60_1_2_4_2_2"/><i><font color="Green" title="E79">A65</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">n1</font>]
 <b>and </b><br/><a NAME="E18:60_1_2_4_2_2"/><i><font color="Green" title="E80">A66</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ]
 <b>from </b><i><a class="ref" href="arytm_3.html#S2" title="ARYTM_3:sch.2">ARYTM_3:sch 2</a>(<a class="txt" href="arytm_2.html#E53"><i><font color="Green" title="E43">Lm33</font></i></a>, <a class="txt" href="arytm_2.html#E54"><i><font color="Green" title="E44">Lm34</font></i></a>, <a class="txt" href="arytm_2.html#E11:60_1_2_4_2_2"><i><font color="Green" title="E75">A61</font></i></a>, <a class="txt" href="arytm_2.html#E13:60_1_2_4_2_2"><i><font color="Green" title="E77">A63</font></i></a>, <a class="txt" href="arytm_2.html#E14:60_1_2_4_2_2"><i><font color="Green" title="E78">A64</font></i></a>);<br/></i><a NAME="E19:60_1_2_4_2_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</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="60_1_2_4_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">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="E1:60_1_2_4_2_2_1"/>
<font color="Maroon" title="c19">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="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 ;<br/>

<a NAME="E2:60_1_2_4_2_2_1"/><b>then </b>
 ex <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 <br/>( <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )
 
;<br/>
<b>hence </b><a NAME="E3:60_1_2_4_2_2_1"/>
<font color="Maroon" title="c19">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="c19">rA</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</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/><div><i><font color="Green" title="E81">A67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_4_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_2_4_2_2_2"/>
<font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">rA</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c20">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="E3:60_1_2_4_2_2_2"/><i><font color="Green" title="E82">A68</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c20">s0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span>
 <b>and </b><br/><a NAME="E4:60_1_2_4_2_2_2"/><i><font color="Green" title="E83">A69</font></i>: 
<font color="Maroon" title="c20">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><i><font color="Green" title="E84">A70</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c20">s0</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">dr</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">x'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E3:60_1_2_4_2_2_2"><i><font color="Green" title="E82">A68</font></i></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></i>
;<br/><a NAME="E6:60_1_2_4_2_2_2"/>
( <font color="Maroon" title="c20">s0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">x'</font> or <font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c20">s0</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c21">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="E8:60_1_2_4_2_2_2"/><i><font color="Green" title="E85">A71</font></i>: 
( ( <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s0</font> &amp; <font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c21">s1</font> ) or ( <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> &amp; <font color="Maroon" title="c20">s0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c21">s1</font> ) )
 ;<br/><a NAME="E9:60_1_2_4_2_2_2"/>
<font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c20">s0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s1</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:60_1_2_4_2_2_2"><i><font color="Green" title="E85">A71</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/><a NAME="E10:60_1_2_4_2_2_2"/><b>then </b><i><font color="Green" title="E86">A72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c20">s0</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">dr</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">x'</font></span>)</span> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">dr</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">x'</font></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></i>;<br/><a NAME="E11:60_1_2_4_2_2_2"/>
<font color="Maroon" title="c14">dr</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c14">dr</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s1</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:60_1_2_4_2_2_2"><i><font color="Green" title="E85">A71</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/><a NAME="E12:60_1_2_4_2_2_2"/><b>then </b><i><font color="Green" title="E87">A73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">dr</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c9">x'</font></span>)</span> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">dr</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s1</font></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s1</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">dr</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c14">dr</font></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s1</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></i>
<br/>.= 
<font color="Maroon" title="c21">s1</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E5:60_1_2_4_2_2"><i><font color="Green" title="E73">A59</font></i></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></i>
;<br/><b>hence </b><a NAME="E14:60_1_2_4_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:60_1_2"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="arytm_2.html#E18:60_1_2"><i><font color="Green" title="E67">A42</font></i></a>, <a class="txt" href="arytm_2.html#E18:60_1_2_4_2_2"><i><font color="Green" title="E80">A66</font></i></a>, <a class="txt" href="arytm_2.html#E4:60_1_2_4_2_2_2"><i><font color="Green" title="E83">A69</font></i></a>, <a class="txt" href="arytm_2.html#E5:60_1_2_4_2_2_2"><i><font color="Green" title="E84">A70</font></i></a>, <a class="txt" href="arytm_2.html#E8:60_1_2_4_2_2_2"><i><font color="Green" title="E85">A71</font></i></a>, <a class="txt" href="arytm_2.html#E10:60_1_2_4_2_2_2"><i><font color="Green" title="E86">A72</font></i></a>, <a class="txt" href="arytm_2.html#E12:60_1_2_4_2_2_2"><i><font color="Green" title="E87">A73</font></i></a>, <a class="txt" href="arytm_2.html#E49"><i><font color="Green" title="E40">Lm30</font></i></a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon" title="c20">s0</font> = <font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span>;<br/><a NAME="E22:60_1_2_4_2_2"/>
<font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <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="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>;<br/><a NAME="E23:60_1_2_4_2_2"/><b>then </b>
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">rA</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E14:60_1_2"><i><font color="Green" title="E65">A31</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">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> <b> such that </b><br/><a NAME="E25:60_1_2_4_2_2"/><i><font color="Green" title="E82">A74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E21:60_1_2_4_2_2"><i><font color="Green" title="E81">A67</font></i></a>, <a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a></i>;<br/><div><i><font color="Green" title="E83">A75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_4_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_2_4_2_2_3"/><i><font color="Green" title="E84">A76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c13">r'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c13">r'</font></span>)</span> = 
<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E3:60_1_2_4_2_2"><i><font color="Green" title="E72">A58</font></i></a>, <a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></i>
<br/>.= 
<font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></i>
;<br/><b>hence </b><a NAME="E3:60_1_2_4_2_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E21:60_1_2_4_2_2"><i><font color="Green" title="E81">A67</font></i></a>, <a class="txt" href="arytm_2.html#E1:60_1_2_4_2_2_3"><i><font color="Green" title="E84">A76</font></i></a></i>;<br/></div><b>end;</b></div><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c13">r'</font></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s'</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c13">r'</font></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">r</font></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s'</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s'</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E3:60_1_2_4_2_2"><i><font color="Green" title="E72">A58</font></i></a>, <a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></i>
<br/>.= 
 <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
<b>by </b><i><a class="txt" href="arytm_2.html#E25:60_1_2_4_2_2"><i><font color="Green" title="E82">A74</font></i></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></i>
;<br/><a NAME="E28:60_1_2_4_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c13">r'</font></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s'</font></span>)</span> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 ;<br/><a NAME="E29:60_1_2_4_2_2"/><b>then </b><i><font color="Green" title="E84">A77</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E26:60_1_2_4_2_2"><i><font color="Green" title="E83">A75</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s'</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">s'</font></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">r</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></i>
<br/>.= 
<font color="Maroon" title="c12">r</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E25:60_1_2_4_2_2"><i><font color="Green" title="E82">A74</font></i></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></i>
;<br/><a NAME="E31:60_1_2_4_2_2"/><b>then </b>
<font color="Maroon" title="c12">r</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">s</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E17:60_1_2_4_2_2"><i><font color="Green" title="E79">A65</font></i></a>, <a class="txt" href="arytm_2.html#E29:60_1_2_4_2_2"><i><font color="Green" title="E84">A77</font></i></a></i>;<br/><b>hence </b><a NAME="E32:60_1_2_4_2_2"/>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c11">B</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E27:60_1_2"/><b>then </b>
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c11">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self" title="ARYTM_2:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E28:60_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E22"><i><font color="Green" title="E17">Lm13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
