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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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">A</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">y'</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="E5:46"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E6:46"/><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>by </b><i><a class="txt" href="arytm_2.html#E14"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="arytm_2.html#E2:46"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E7:46"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</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="E9:46"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T49">SUBSET_1:49</a></i>;<br/>
<b>consider </b><font color="Maroon">n</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="E11:46"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>and </b><br/><a NAME="E12:46"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">r</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">n</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T103">ARYTM_3:103</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>;<br/>
<a NAME="E13:46"/><i><font color="Green">E39</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 
<b>by </b><i>, , <a class="txt" href="arytm_2.html#E3:46"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">A0</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="E15:46"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">A0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E16:46"/><i><font color="Green">E73</font></i>: 
 <a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">A0</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/>
<a NAME="E17:46"/>
<a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T54">ARYTM_3:54</a></i>;<br/>
<a NAME="E18:46"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="arytm_3.html#K13">{}</a> ]
 
<b>by </b><i><a class="txt" href="arytm_2.html#E18"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="arytm_2.html#E3:46"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">n0</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="E20:46"/>
<font color="Maroon">n0</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>and </b><br/><a NAME="E21:46"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n0</font>]
 <b>and </b><br/><a NAME="E22:46"/><i><font color="Green">E76</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n0</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K14">one</a> ]
 <b>from </b><i><a class="ref" href="arytm_3.html#S2">ARYTM_3:sch 2</a>(<a class="txt" href="arytm_2.html#E10"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="arytm_2.html#T2" target="_self">Th2</a>, <a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="arytm_2.html#E17"><i><font color="Green">Lemma53</font></i></a>);<br/></i>
<i><font color="Green">E77</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">y'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">y'</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
;<br/>
<b>thus </b><a NAME="E24:46"/>
contradiction
 <b>by </b><i>, <a class="ref" href="arytm_2.html#D4" target="_self">Def4</a>, <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/>


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