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<b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">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><a NAME="E1:62"/><i><font color="Green" title="E58">A1</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">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#K5" title="ARYTM_3:func.5">RAT+</a>  : <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#K12" title="ARYTM_3:func.12">one</a>  </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E2:62"/>
<font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
  <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="62_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:62_1"/>
<font color="Maroon" title="c6">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="c5">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">r</font>, <font color="Maroon" title="c8">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E3:62_1"/><i><font color="Green" title="E59">A2</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">s</font>
 <b>and </b><br/><a NAME="E4:62_1"/><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c7">r</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="E5:62_1"/><i><font color="Green" title="E61">A4</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ;<br/>
<a NAME="E6:62_1"/>
 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#K5" title="ARYTM_3:func.5">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">r</font> &amp; <font color="Olive" title="b1">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="arytm_2.html#E4:62_1"><i><font color="Green" title="E60">A3</font></i></a></i>;<br/>
<a NAME="E7:62_1"/><b>then </b>
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">s</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="E8:62_1"/><b>then </b>
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">s</font>
 
<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="E9:62_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:62_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="arytm_2.html#E5:62_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E21">Lm17</font></i></a></i>;<br/>


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

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

<b>assume </b><a NAME="E3:62"/><i><font color="Green" title="E59">A5</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c6">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c8">s1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E6:62"/><i><font color="Green" title="E60">A6</font></i>: 
<font color="Maroon" title="c8">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E7:62"/><i><font color="Green" title="E61">A7</font></i>: 
<font color="Maroon" title="c7">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">s1</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:62"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="arytm_2.html#E17"><i><font color="Green" title="E11">Lm8</font></i></a></i>;<br/>
<a NAME="E8:62"/>
<font color="Maroon" title="c8">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E7:62"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">s2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E10:62"/><i><font color="Green" title="E62">A8</font></i>: 
<font color="Maroon" title="c8">s1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a></i>;<br/>
<a NAME="E11:62"/>
<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s2</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">s1</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E10:62"><i><font color="Green" title="E62">A8</font></i></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></i>;<br/>
<a NAME="E12:62"/><b>then </b><i><font color="Green" title="E63">A9</font></i>: 
<font color="Maroon" title="c9">s2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E7:62"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="arytm_3.html#T91" title="ARYTM_3:th.91">ARYTM_3:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_2"/>
<font color="Maroon" title="c9">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 ;<br/><a NAME="E2:62_2"/><b>then </b>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:62"><i><font color="Green" title="E62">A8</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="E3:62_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E7:62"><i><font color="Green" title="E61">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:62"/><b>then </b>
<font color="Maroon" title="c9">s2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E12:62"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/>
<a NAME="E15:62"/><b>then </b>
<font color="Maroon" title="c9">s2</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#E1:62"><i><font color="Green" title="E58">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:62"/>
<font color="Maroon" title="c6">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="c5">B</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:62"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="arytm_2.html#E10:62"><i><font color="Green" title="E62">A8</font></i></a></i>;<br/>


</div>
