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

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">B</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">DIF</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> 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>  :  ex <font color="Olive" title="b2">r</font>, <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/>( not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> ) </span>}</span> ;<br/>
<a NAME="E4:48"/>
not <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E2:48"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="arytm_2.html#E3:48"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">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="E6:48"/><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c6">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E7:48"/><i><font color="Green" title="E52">A5</font></i>: 
not <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<a NAME="E8:48"/>
<font color="Maroon" title="c6">s1</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s1</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T92" title="ARYTM_3:th.92">ARYTM_3:92</a></i>;<br/>
<a NAME="E9:48"/><b>then </b><i><font color="Green" title="E53">A6</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">t</font> 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>  :  ex <font color="Olive" title="b2">r</font>, <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/>( not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E6:48"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="arytm_2.html#E7:48"><i><font color="Green" title="E52">A5</font></i></a></i>;<br/>
<a NAME="E10:48"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> 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>  :  ex <font color="Olive" title="b2">r</font>, <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/>( not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</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="48_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="E1:48_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">t</font> 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>  :  ex <font color="Olive" title="b2">r</font>, <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/>( not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:48_1"/><b>then </b>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Olive" title="b1">t</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">r</font>, <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/>( not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:48_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">DIF</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> 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>  :  ex <font color="Olive" title="b2">r</font>, <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/>( not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> ) </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#E9:48"><i><font color="Green" title="E53">A6</font></i></a></i>;<br/>
<a NAME="E12:48"/>
for <font color="Olive" title="b1">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>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">DIF</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">t</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">DIF</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">DIF</font> &amp; <font color="Olive" title="b1">t</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="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">t</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="E1:48_2"/>
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">DIF</font>
 ;<br/>

<a NAME="E2:48_2"/><b>then </b>
 ex <font color="Olive" title="b1">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/>( <font color="Olive" title="b1">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> &amp;  ex <font color="Olive" title="b2">r</font>, <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/>( not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> ) )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c9">r0</font>, <font color="Maroon" title="c10">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="E4:48_2"/><i><font color="Green" title="E54">A7</font></i>: 
not <font color="Maroon" title="c9">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E5:48_2"/><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon" title="c10">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E6:48_2"/><i><font color="Green" title="E56">A9</font></i>: 
<font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s0</font>
 ;<br/>
<b>thus </b><a NAME="E7:48_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="c8">t</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">DIF</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_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="E1:48_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="c8">t</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">p</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="E3:48_2_1"/><i><font color="Green" title="E57">A10</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></i>;<br/>
<a NAME="E4:48_2_1"/><i><font color="Green" title="E58">A11</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">t</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E3:48_2_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="arytm_3.html#T85" title="ARYTM_3:th.85">ARYTM_3:85</a></i>;<br/>
<a NAME="E5:48_2_1"/>
<font color="Maroon" title="c8">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E6:48_2"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="arytm_3.html#T85" title="ARYTM_3:th.85">ARYTM_3:85</a></i>;<br/>
<a NAME="E6:48_2_1"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E4:48_2_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">q</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="E8:48_2_1"/><i><font color="Green" title="E59">A12</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></i>;<br/>
<a NAME="E9:48_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">p</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E6:48_2"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="arytm_2.html#E3:48_2_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="arytm_2.html#E8:48_2_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></i>;<br/>
<a NAME="E10:48_2_1"/><b>then </b><i><font color="Green" title="E60">A13</font></i>: 
<font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T69" title="ARYTM_3:th.69">ARYTM_3:69</a></i>;<br/>
<a NAME="E11:48_2_1"/>
<font color="Maroon" title="c13">q</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E8:48_2_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="arytm_3.html#T85" title="ARYTM_3:th.85">ARYTM_3:85</a></i>;<br/>
<a NAME="E12:48_2_1"/><b>then </b>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:48_2"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E22">Lm17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:48_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">DIF</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:48_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="arytm_2.html#E10:48_2_1"><i><font color="Green" title="E60">A13</font></i></a></i>;<br/>


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

<b>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="E9:48_2"/><i><font color="Green" title="E57">A14</font></i>: 
<font color="Maroon" title="c11">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E10:48_2"/><i><font color="Green" title="E58">A15</font></i>: 
<font color="Maroon" title="c10">s0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c11">s1</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:48_2"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="arytm_2.html#E17"><i><font color="Green" title="E12">Lm8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">q</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="E12:48_2"/><i><font color="Green" title="E59">A16</font></i>: 
<font color="Maroon" title="c10">s0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s1</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:48_2"><i><font color="Green" title="E58">A15</font></i></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">t</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">q</font>
;<br/>

<a NAME="E13:48_2"/><i><font color="Green" title="E60">A17</font></i>: 
<font color="Maroon" title="c9">r0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s1</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E6:48_2"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="arytm_2.html#E12:48_2"><i><font color="Green" title="E59">A16</font></i></a>, <a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></i>;<br/>
<b>hence </b><a NAME="E14:48_2"/>
<font color="Maroon" title="c8">t</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">DIF</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:48_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="arytm_2.html#E9:48_2"><i><font color="Green" title="E57">A14</font></i></a></i>;<br/>

<a NAME="E15:48_2"/><i><font color="Green" title="E61">A18</font></i>: 
<font color="Maroon" title="c8">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">t</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">q</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T85" title="ARYTM_3:th.85">ARYTM_3:85</a></i>;<br/>
<a NAME="E16:48_2"/>
<font color="Maroon" title="c8">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">t</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">q</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E6:48_2"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="arytm_2.html#E10:48_2"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="arytm_2.html#E13:48_2"><i><font color="Green" title="E60">A17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:48_2"/>
<font color="Maroon" title="c8">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c12">q</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E15:48_2"><i><font color="Green" title="E61">A18</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="E13:48"/><b>then </b><i><font color="Green" title="E54">A19</font></i>: 
<font color="Maroon" title="c7">DIF</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/>
<a NAME="E14:48"/>
<font color="Maroon" title="c4">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> 
 
<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="c8">s2</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="E16:48"/><i><font color="Green" title="E55">A20</font></i>: 
not <font color="Maroon" title="c8">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T49" title="SUBSET_1:th.49">SUBSET_1:49</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3"/>
<font color="Maroon" title="c8">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">DIF</font>
 ;<br/><a NAME="E2:48_3"/><b>then </b>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s2</font> &amp;  ex <font color="Olive" title="b2">r</font>, <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/>( not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> ) )
 ;<br/><b>then consider </b><font color="Maroon" title="c9">r</font>, <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="E4:48_3"/>
not <font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E5:48_3"/><i><font color="Green" title="E56">A21</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E6:48_3"/><i><font color="Green" title="E57">A22</font></i>: 
<font color="Maroon" title="c9">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font>
 ;<br/><a NAME="E7:48_3"/>
<font color="Maroon" title="c8">s2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">t</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:48_3"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="arytm_3.html#T85" title="ARYTM_3:th.85">ARYTM_3:85</a></i>;<br/><b>hence </b><a NAME="E8:48_3"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E16:48"><i><font color="Green" title="E55">A20</font></i></a>, <a class="txt" href="arytm_2.html#E5:48_3"><i><font color="Green" title="E56">A21</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E22">Lm17</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:48"/><b>then </b>
<font color="Maroon" title="c7">DIF</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">DIF</font> = <font color="Maroon" title="c7">DIF</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#E13:48"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">DIF</font>
;<br/>

<b>set </b><font color="Maroon" title="c10">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">r</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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">DIF</font> ) </span>}</span> ;<br/>
<a NAME="E20:48"/>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">r</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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">DIF</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_4"/>
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">r</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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">DIF</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="48_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="E1:48_4_1"/><i><font color="Green" title="E56">A23</font></i>: 
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">y'</font> = <font color="Maroon" title="c11">e</font> as    <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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_4_1_1"/>
( <font color="Maroon" title="c12">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> or  not <font color="Maroon" title="c12">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_1"><b>suppose </b></a><a NAME="E1:48_4_1_1_1"/><i><font color="Green" title="E57">A24</font></i>: 
<font color="Maroon" title="c12">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><div class="add"><a NAME="E2:48_4_1_1_1"/>
<font color="Maroon" title="c12">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T92" title="ARYTM_3:th.92">ARYTM_3:92</a></i>;<br/><b>hence </b><a NAME="E3:48_4_1_1_1"/>
<font color="Maroon" title="c11">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#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">r</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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">DIF</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E9:48"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="arytm_2.html#E1:48_4_1_1_1"><i><font color="Green" title="E57">A24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_2"><b>suppose </b></a><a NAME="E1:48_4_1_1_2"/><i><font color="Green" title="E57">A25</font></i>: 
not <font color="Maroon" title="c12">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c13">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="E3:48_4_1_1_2"/><i><font color="Green" title="E58">A26</font></i>: 
<font color="Maroon" title="c13">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E4:48_4_1_1_2"/><i><font color="Green" title="E59">A27</font></i>: 
<font color="Maroon" title="c12">y'</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:48_4_1"><i><font color="Green" title="E56">A23</font></i></a>, <a class="txt" href="arytm_2.html#E17"><i><font color="Green" title="E12">Lm8</font></i></a></i>;<br/><a NAME="E5:48_4_1_1_2"/><i><font color="Green" title="E60">A28</font></i>: 
not <font color="Maroon" title="c13">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:48_4_1_1_2"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="arytm_2.html#E4:48_4_1_1_2"><i><font color="Green" title="E59">A27</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E22">Lm17</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c14">Z</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</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="c3">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s0</font> ) </span>}</span> ;<br/><a NAME="E6:48_4_1_1_2"/><i><font color="Green" title="E61">A29</font></i>: 
<font color="Maroon" title="c13">s0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T92" title="ARYTM_3:th.92">ARYTM_3:92</a></i>;<br/><a NAME="E7:48_4_1_1_2"/>
for <font color="Olive" title="b1">r2</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">r2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c13">s0</font> holds <br/> ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">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>  st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">r</font> where <font color="Olive" title="b4">r</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="b5">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="b5">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b5">x'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s0</font> ) </span>}</span>  &amp; <font color="Olive" title="b1">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b3">t</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">r2</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="E1:48_4_1_1_2_1"/><i><font color="Green" title="E62">A30</font></i>: 
<font color="Maroon" title="c15">r2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c13">s0</font>
 ;<br/>

<a NAME="E2:48_4_1_1_2_1"/><b>then </b><i><font color="Green" title="E63">A31</font></i>: 
( <font color="Maroon" title="c15">r2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c13">s0</font> &amp; <font color="Maroon" title="c15">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">s0</font> )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_4_1_1_2_1_1"/>
( <font color="Maroon" title="c15">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> or  not <font color="Maroon" title="c15">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_4_1_1_2_1_1_1"/><i><font color="Green" title="E64">A32</font></i>: 
<font color="Maroon" title="c15">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c15">r2</font>
;<br/><b>take </b>
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
;<br/><b>thus </b><a NAME="E2:48_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:48_4_1_1_2_1_1_1"><i><font color="Green" title="E64">A32</font></i></a></i>;<br/><b>thus </b><a NAME="E3:48_4_1_1_2_1_1_1"/>
 <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">r</font> where <font color="Olive" title="b1">r</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="c3">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s0</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:48_4_1_1_2"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="arytm_2.html#E6:48_4_1_1_2"><i><font color="Green" title="E61">A29</font></i></a></i>;<br/><b>thus </b><a NAME="E4:48_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r2</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T92" title="ARYTM_3:th.92">ARYTM_3:92</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_4_1_1_2_1_1_2"/><i><font color="Green" title="E64">A33</font></i>: 
not <font color="Maroon" title="c15">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c16">q</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="E3:48_4_1_1_2_1_1_2"/><i><font color="Green" title="E65">A34</font></i>: 
<font color="Maroon" title="c15">r2</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:48_4_1_1_2_1"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></i>;<br/><a NAME="E4:48_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c16">q</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#E2:48_4_1_1_2_1"><i><font color="Green" title="E63">A31</font></i></a>, <a class="txt" href="arytm_2.html#E3:48_4_1_1_2_1_1_2"><i><font color="Green" title="E65">A34</font></i></a>, <a class="ref" href="arytm_3.html#T92" title="ARYTM_3:th.92">ARYTM_3:92</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">n</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="E6:48_4_1_1_2_1_1_2"/><i><font color="Green" title="E66">A35</font></i>: 
<font color="Maroon" title="c17">n</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="E7:48_4_1_1_2_1_1_2"/><i><font color="Green" title="E67">A36</font></i>: 
<font color="Maroon" title="c13">s0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c17">n</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font>
 <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> <font color="Maroon" title="c16">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>;<br/><a NAME="E8:48_4_1_1_2_1_1_2"/><i><font color="Green" title="E68">A37</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">n</font>]
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:48_4_1_1_2"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="arytm_2.html#E7:48_4_1_1_2_1_1_2"><i><font color="Green" title="E67">A36</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E22">Lm17</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c18">y0</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="E10:48_4_1_1_2_1_1_2"/><i><font color="Green" title="E69">A38</font></i>: 
<font color="Maroon" title="c18">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:48"><i><font color="Green" title="E48">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="E11:48_4_1_1_2_1_1_2"/>
<a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>;<br/><a NAME="E12:48_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E70">A39</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#E10:48_4_1_1_2_1_1_2"><i><font color="Green" title="E69">A38</font></i></a>, <a class="txt" href="arytm_2.html#E28"><i><font color="Green" title="E23">Lm18</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c19">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="E14:48_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c19">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="E15:48_4_1_1_2_1_1_2"/><i><font color="Green" title="E71">A40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">n0</font>]
 <b>and </b><br/><a NAME="E16:48_4_1_1_2_1_1_2"/><i><font color="Green" title="E72">A41</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">n0</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#E6:48_4_1_1_2_1_1_2"><i><font color="Green" title="E66">A35</font></i></a>, <a class="txt" href="arytm_2.html#E12:48_4_1_1_2_1_1_2"><i><font color="Green" title="E70">A39</font></i></a>, <a class="txt" href="arytm_2.html#E8:48_4_1_1_2_1_1_2"><i><font color="Green" title="E68">A37</font></i></a>);<br/></i><i><font color="Green" title="E73">A42</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c19">n0</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> <font color="Maroon" title="c16">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">n0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font></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> <font color="Maroon" title="c16">q</font></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="c19">n0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c16">q</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></i>
;<br/><a NAME="E18:48_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c19">n0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c15">r2</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:48_4_1_1_2_1_1_2"><i><font color="Green" title="E64">A33</font></i></a>, <a class="txt" href="arytm_2.html#E15:48_4_1_1_2_1_1_2"><i><font color="Green" title="E71">A40</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E22">Lm17</font></i></a></i>;<br/><a NAME="E19:48_4_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">n0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c16">q</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:48_4_1_1_2_1_1_2"><i><font color="Green" title="E65">A34</font></i></a>, <a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">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="E21:48_4_1_1_2_1_1_2"/><i><font color="Green" title="E74">A43</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">n0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></i>;<br/><b>take </b>
<font color="Maroon" title="c19">n0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font>
;<br/><b>take </b>
<font color="Maroon" title="c20">t</font>
;<br/><b>thus </b><a NAME="E22:48_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c19">n0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E15:48_4_1_1_2_1_1_2"><i><font color="Green" title="E71">A40</font></i></a></i>;<br/><b>thus </b><a NAME="E23:48_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</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="c3">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s0</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E16:48_4_1_1_2_1_1_2"><i><font color="Green" title="E72">A41</font></i></a>, <a class="txt" href="arytm_2.html#E17:48_4_1_1_2_1_1_2"><i><font color="Green" title="E73">A42</font></i></a>, <a class="txt" href="arytm_2.html#E21:48_4_1_1_2_1_1_2"><i><font color="Green" title="E74">A43</font></i></a></i>;<br/><a NAME="E24:48_4_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">n0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c20">t</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r2</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c16">q</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:48_4_1_1_2_1_1_2"><i><font color="Green" title="E65">A34</font></i></a>, <a class="txt" href="arytm_2.html#E21:48_4_1_1_2_1_1_2"><i><font color="Green" title="E74">A43</font></i></a>, <a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></i>;<br/><b>hence </b><a NAME="E25:48_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c15">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n0</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c20">t</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T69" title="ARYTM_3:th.69">ARYTM_3:69</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c15">s</font>, <font color="Maroon" title="c16">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="E9:48_4_1_1_2"/><i><font color="Green" title="E62">A44</font></i>: 
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E10:48_4_1_1_2"/><i><font color="Green" title="E63">A45</font></i>: 
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</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="c3">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s0</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E11:48_4_1_1_2"/><i><font color="Green" title="E64">A46</font></i>: 
<font color="Maroon" title="c12">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c16">t</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:48_4_1_1_2"><i><font color="Green" title="E59">A27</font></i></a></i>;<br/><a NAME="E12:48_4_1_1_2"/>
 ex <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 <br/>( <font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</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="c3">A</font> &amp; <font color="Olive" title="b2">x'</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s0</font> ) )
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:48_4_1_1_2"><i><font color="Green" title="E63">A45</font></i></a></i>;<br/><a NAME="E13:48_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">DIF</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:48_4_1_1_2"><i><font color="Green" title="E58">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E14:48_4_1_1_2"/>
<font color="Maroon" title="c11">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#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">r</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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">DIF</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E9:48_4_1_1_2"><i><font color="Green" title="E62">A44</font></i></a>, <a class="txt" href="arytm_2.html#E11:48_4_1_1_2"><i><font color="Green" title="E64">A46</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c11">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="E2:48_4"/>
<font color="Maroon" title="c11">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#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">r</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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">DIF</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">s3</font>, <font color="Maroon" title="c13">t3</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="E4:48_4"/><i><font color="Green" title="E56">A47</font></i>: 
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s3</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c13">t3</font>
 <b>and </b><br/><a NAME="E5:48_4"/><i><font color="Green" title="E57">A48</font></i>: 
<font color="Maroon" title="c12">s3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E6:48_4"/><i><font color="Green" title="E58">A49</font></i>: 
<font color="Maroon" title="c13">t3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">DIF</font>
 ;<br/>
<a NAME="E7:48_4"/>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c13">t3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> &amp;  ex <font color="Olive" title="b2">r</font>, <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/>( not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> ) )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E6:48_4"><i><font color="Green" title="E58">A49</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">r4</font>, <font color="Maroon" title="c15">s4</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="E9:48_4"/><i><font color="Green" title="E59">A50</font></i>: 
not <font color="Maroon" title="c14">r4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E10:48_4"/><i><font color="Green" title="E60">A51</font></i>: 
<font color="Maroon" title="c15">s4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E11:48_4"/><i><font color="Green" title="E61">A52</font></i>: 
<font color="Maroon" title="c14">r4</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c13">t3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s4</font>
 ;<br/>
<a NAME="E12:48_4"/>
<font color="Maroon" title="c12">s3</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c14">r4</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:48_4"><i><font color="Green" title="E57">A48</font></i></a>, <a class="txt" href="arytm_2.html#E9:48_4"><i><font color="Green" title="E59">A50</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E22">Lm17</font></i></a></i>;<br/>
<a NAME="E13:48_4"/><b>then </b>
<font color="Maroon" title="c12">s3</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c13">t3</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c15">s4</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E11:48_4"><i><font color="Green" title="E61">A52</font></i></a>, <a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></i>;<br/>
<b>hence </b><a NAME="E14:48_4"/>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:48_4"><i><font color="Green" title="E56">A47</font></i></a>, <a class="txt" href="arytm_2.html#E10:48_4"><i><font color="Green" title="E60">A51</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E22">Lm17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:48"/>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c9">DIF</font>
 ;<br/>


</div>
