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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="E1:36"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K5" 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="E2:36"/><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> &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> 
 
;<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="E4:36"/><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#K11">+</a> <font color="Maroon">s0</font>
 <b>and </b><br/><a NAME="E5:36"/><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="E6:36"/><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#K5" target="_self">+</a> <font color="Maroon">C</font>
 ;<br/>
<a NAME="E7:36"/>
<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#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">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#E12:36"><i><font color="Green">E37</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="E9:36"/><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#K11">+</a> <font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E10:36"/><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="E11:36"/><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="E12:36"/><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#K11">+</a> <font color="Maroon">r1</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5"><i><font color="Green">Lemma35</font></i></a>, , <a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>;<br/>
<a NAME="E13:36"/>
<font color="Maroon">r0</font> <a href="arytm_3.html#K11">+</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#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> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:36"/><b>then </b>
<font color="Maroon">r0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">B</font>
 
;<br/>
<a NAME="E15:36"/><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#K5" 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="ref" href="arytm_2.html#T1" target="_self">Th1</a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:36"/>
<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#K5" target="_self">+</a> <font color="Maroon">B</font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">C</font>
 ;<br/>


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