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<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  implies <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c5">y'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E5:46"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c5">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E6:46"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c5">y'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:46"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="arytm_2.html#E25"><i><font color="Green" title="E19">Lm15</font></i></a></i>;<br/>
<a NAME="E7:46"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E9:46"/><i><font color="Green" title="E49">A6</font></i>: 
not <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T49" title="SUBSET_1:th.49">SUBSET_1:49</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E11:46"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>and </b><br/><a NAME="E12:46"/><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c7">n</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">y'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:46"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="arytm_3.html#T103" title="ARYTM_3:th.103">ARYTM_3:103</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> ] <b>means </b>$1 <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>;<br/>
<a NAME="E13:46"/><i><font color="Green" title="E52">A9</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>]
 
<b>by </b><i><a class="txt" href="arytm_2.html#E9:46"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="arytm_2.html#E12:46"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green" title="E20">Lm16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">A0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E15:46"/><i><font color="Green" title="E53">A10</font></i>: 
<font color="Maroon" title="c8">A0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:46"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<a NAME="E16:46"/>
<a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>;<br/>
<a NAME="E17:46"/><b>then </b><i><font color="Green" title="E54">A11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> ]
 
<b>by </b><i><a class="txt" href="arytm_2.html#E15:46"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green" title="E20">Lm16</font></i></a>, <a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">n0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E19:46"/>
<font color="Maroon" title="c9">n0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>and </b><br/><a NAME="E20:46"/><i><font color="Green" title="E55">A12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n0</font>]
 <b>and </b><br/><a NAME="E21:46"/><i><font color="Green" title="E56">A13</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> ]
 <b>from </b><i><a class="ref" href="arytm_3.html#S1" title="ARYTM_3:sch.1">ARYTM_3:sch 1</a>(<a class="txt" href="arytm_2.html#E52"><i><font color="Green" title="E41">Lm32</font></i></a>, <a class="txt" href="arytm_2.html#E53"><i><font color="Green" title="E42">Lm33</font></i></a>, <a class="txt" href="arytm_2.html#E11:46"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="arytm_2.html#E17:46"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="arytm_2.html#E13:46"><i><font color="Green" title="E52">A9</font></i></a>);<br/></i>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">y'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">y'</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">y'</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c5">y'</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></i>
;<br/>
<b>hence </b><a NAME="E23:46"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:46"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="arytm_2.html#E5:46"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="arytm_2.html#E20:46"><i><font color="Green" title="E55">A12</font></i></a>, <a class="txt" href="arytm_2.html#E21:46"><i><font color="Green" title="E56">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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