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

<b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> , <font color="Maroon">B</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> , <font color="Maroon">C</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> ;<br/>

<b>thus </b><a NAME="E1:57"/>
<font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">B</font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">C</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<a NAME="E2:57_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">r0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">v0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E4:57_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">v0</font>
 <b>and </b><br/><a NAME="E5:57_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:57_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">v0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font>
 ;<br/>
<a NAME="E7:57_1"/>
<font color="Maroon">v0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E21"><i><font color="Green">Lemma57</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">s0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">t0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E9:57_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">v0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0</font>
 <b>and </b><br/><a NAME="E10:57_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E11:57_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>
<a NAME="E12:57_1"/>
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a>, </i>;<br/>
<a NAME="E13:57_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">B</font>
 
;<br/>
<a NAME="E14:57_1"/>
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a>, </i>;<br/>
<a NAME="E15:57_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">C</font>
 
;<br/>
<a NAME="E16:57_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>;<br/>
<a NAME="E17:57_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">C</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#D5" target="_self">Def5</a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:57_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">B</font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>


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

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

<b>assume </b><a NAME="E2:57"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">B</font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<a NAME="E3:57"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">C</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">s1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">t1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E5:57"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t1</font>
 <b>and </b><br/><a NAME="E6:57"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E7:57"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">C</font>
 ;<br/>
<a NAME="E8:57"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">r0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">s0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E10:57"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0</font>
 <b>and </b><br/><a NAME="E11:57"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E12:57"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>
<a NAME="E13:57"/>
<font color="Maroon">t1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E21"><i><font color="Green">Lemma57</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">r0'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">t0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E15:57"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0'</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>and </b><br/><a NAME="E16:57"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">r0'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E17:57"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2"/>
( <font color="Maroon">r0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r0'</font> or <font color="Maroon">r0'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1"><b>suppose </b></a><a NAME="E1:57_2_1"/>
<font color="Maroon">r0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r0'</font>
 ;<br/><div class="add"><a NAME="E2:57_2_1"/><b>then </b>
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r0'</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/><b>then consider </b><font color="Maroon">s0'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E4:57_2_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0'</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0'</font>
 <b>and </b><br/><a NAME="E5:57_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">s0'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T87">ARYTM_3:87</a></i>;<br/><a NAME="E6:57_2_1"/>
<font color="Maroon">s0'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E6:57"><i><font color="Green">E65</font></i></a>, </i>;<br/><a NAME="E7:57_2_1"/><b>then </b>
<font color="Maroon">s0'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a></i>;<br/><a NAME="E8:57_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">s0'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E9:57_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0'</font> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">s0'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a>, , <a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>;<br/><a NAME="E10:57_2_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green">Lemma66</font></i></a></i>;<br/><b>hence </b><a NAME="E11:57_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2"><b>suppose </b></a><a NAME="E1:57_2_2"/>
<font color="Maroon">r0'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r0</font>
 ;<br/><div class="add"><a NAME="E2:57_2_2"/><b>then </b>
<font color="Maroon">r0'</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/><b>then consider </b><font color="Maroon">t0'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E4:57_2_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">r0'</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0'</font>
 <b>and </b><br/><a NAME="E5:57_2_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">t0'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T87">ARYTM_3:87</a></i>;<br/><a NAME="E6:57_2_2"/>
<font color="Maroon">t0'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="arytm_2.html#E6:57"><i><font color="Green">E65</font></i></a>, </i>;<br/><a NAME="E7:57_2_2"/><b>then </b>
<font color="Maroon">s0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E8:57_2_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">s0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E9:57_2_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0'</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a>, , <a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>;<br/><a NAME="E10:57_2_2"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font> ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green">Lemma66</font></i></a></i>;<br/><b>hence </b><a NAME="E11:57_2_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
