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<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>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="E1:53"/>
<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#K6" target="_self">*'</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<a NAME="E2:53"/><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#K6" 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">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="E4:53"/><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">s0</font>
 <b>and </b><br/><a NAME="E5:53"/><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="E6:53"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">C</font>
 ;<br/>
<a NAME="E7:53"/>
<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">B</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#E18"><i><font color="Green">Lemma54</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">s1</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="E9:53"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E10:53"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E11:53"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>
<a NAME="E12:53"/><i><font color="Green">E37</font></i>: 
<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">r1</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, <a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>;<br/>
<a NAME="E13:53"/>
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r1</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#E17"><i><font color="Green">Lemma53</font></i></a>, <a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a></i>;<br/>
<a NAME="E14:53"/><b>then </b>
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r1</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="E15:53"/><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> <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">C</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E21"><i><font color="Green">Lemma57</font></i></a>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:53"/>
<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#K6" target="_self">*'</a> <font color="Maroon">C</font>
 ;<br/>


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