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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E3:46"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:46"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
 ex <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">A0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="subset_1.html#T4" title="SUBSET_1:th.4">SUBSET_1:4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">y9</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> <span class="kw"> such that </span><br/><a NAME="E6:46"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c5">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E7:46"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<font color="Maroon" title="c5">y9</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="arytm_2.html#E25"><span class="lab"><font color="Green" title="E19">Lm15</font></span></a></span>;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span>$1 <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>;<br/>
<a NAME="E8:46"/>
<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">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T48" title="ARYTM_3:th.48">ARYTM_3:48</a></span>;<br/>
<a NAME="E9:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="arytm_2.html#E26"><span class="lab"><font color="Green" title="E20">Lm16</font></span></a>, <a class="ref" href="arytm_3.html#T64" title="ARYTM_3:th.64">ARYTM_3:64</a></span>;<br/>
<a NAME="E10:46"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E12:46"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
 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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T28" title="SUBSET_1:th.28">SUBSET_1:28</a></span>;<br/>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E14:46"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 <span class="kw">and </span><br/><a NAME="E15:46"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<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">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="arytm_3.html#T95" title="ARYTM_3:th.95">ARYTM_3:95</a></span>;<br/>
<a NAME="E16:46"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E15:46"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="arytm_2.html#E26"><span class="lab"><font color="Green" title="E20">Lm16</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">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> <span class="kw"> such that </span><br/><a NAME="E18:46"/>
<font color="Maroon" title="c8">n0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 <span class="kw">and </span><br/><a NAME="E19:46"/><span class="lab"><font color="Green" title="E55">A12</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n0</font>] &amp;  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">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 class="kw">from</span> <span class="lab"><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#E51"><span class="lab"><font color="Green" title="E41">Lm32</font></span></a>, <a class="txt" href="arytm_2.html#E52"><span class="lab"><font color="Green" title="E42">Lm33</font></span></a>, <a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="txt" href="#E16:46"><span class="lab"><font color="Green" title="E54">A11</font></span></a>);<br/></span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">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">y9</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">n0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">y9</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">y9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">n0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">y9</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c5">y9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T53" title="ARYTM_3:th.53">ARYTM_3:53</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E21:46"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E19:46"><span class="lab"><font color="Green" title="E55">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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