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





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  &amp; <font color="Maroon">z</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="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green">E20</font></i>: 
<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><font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_3"><b>suppose </b></a><a NAME="E1:39_1_3"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:39_1_4"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>and </b><br/><a NAME="E2:39_1_4"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>and </b><br/><a NAME="E3:39_1_4"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_4_1"/>
 <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">y</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">z</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><a NAME="E2:39_1_4_1"/><b>then </b>
<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="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">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E3:39_1_4_1"/><b>then </b>
(  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:39_1_4_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="arytm_2.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_4_2"/>
 <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> 
 ;<br/><a NAME="E2:39_1_4_2"/><b>then </b>
<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="E3:39_1_4_2"/><b>then </b>
(  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:39_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:39_1_4"><i><font color="Green">E38</font></i></a>, , </i>;<br/></div><b>end;</b></div><b>thus </b><font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <span class="p3">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_2.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
 <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> <span class="p3">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <span class="p5">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E5:39_1_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma41</font></i></a>, </i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <span class="p3">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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="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">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E5"><i><font color="Green">Lemma35</font></i></a></i>
<br/>.= 
 <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> <span class="p3">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></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">z</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self">Th1</a>, <a class="txt" href="arytm_2.html#E10"><i><font color="Green">Lemma42</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E5:39_1_4"><i><font color="Green">E38</font></i></a>, , </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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