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<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E4"><i><font color="Green">Lemma34</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>assume </b><a NAME="E2:35_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><a NAME="E3:35_1_2"/><b>then </b>
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E4"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="arytm_2.html#E5"><i><font color="Green">Lemma35</font></i></a>, </i>;<br/><a NAME="E4:35_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E5:35_1_2"/>
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i>, </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> <b> such that </b><br/><a NAME="E7:35_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><a NAME="E8:35_1_2"/>
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E5"><i><font color="Green">Lemma35</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> <b> such that </b><br/><a NAME="E10:35_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><a NAME="E11:35_1_2"/>
<font color="Maroon">r0</font> <a href="arytm_3.html#K11">+</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#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>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E12:35_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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