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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">w</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>take </b>
<font color="Maroon">w</font>
;<br/><b>thus </b><a NAME="E3:27_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">w</font> &amp; <font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">w</font> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E4:27_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font> )
 ;<br/><b>hence </b><a NAME="E5:27_1_1_1_1"/>
<font color="Maroon">w</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_2"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">k</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">k</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">k</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> ex <font color="Olive">m</font> being <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> &amp; <font color="Maroon">k</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Olive">m</font> &amp; <font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Olive">m</font> &amp; <font color="Maroon">k</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">m</font> );<br/><a NAME="E3:27_1_1_1_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E4:27_1_1_1_2"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E13:27_1_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E5:27_1_1_1_2"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E6:27_1_1_1_2"/><b>then </b>
1 <a href="ordinal1.html#R1">c=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a>, <a class="txt" href="arytm_3.html#E11:27_1_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E7:27_1_1_1_2"/><b>then </b>
( <font color="Maroon">k</font> <a href="arytm_3.html#K3" target="_self">*^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="arytm_3.html#K3" target="_self">*^</a> 1 <a href="ordinal1.html#R1">c=</a> <font color="Maroon">k</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a>, <a class="ref" href="ordinal2.html#T58">ORDINAL2:58</a></i>;<br/><a NAME="E8:27_1_1_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>]
 <b>by </b><i><a class="txt" href="arytm_3.html#E16:27_1_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/><b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E10:27_1_1_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A</font>]
 <b>and </b><br/><a NAME="E11:27_1_1_1_2"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>();<br/></i><b>consider </b><font color="Maroon">l</font> being  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E13:27_1_1_1_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">k</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">l</font> &amp; <font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">l</font> &amp; <font color="Maroon">k</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">l</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>take </b>
<font color="Maroon">l</font>
;<br/><b>thus </b><a NAME="E15:27_1_1_1_2"/>
( <font color="Maroon">k</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">l</font> &amp; <font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">l</font> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E16:27_1_1_1_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">k</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_2_1"><b>now </b></a><div class="add"><a NAME="E1:27_1_1_1_2_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E2:27_1_1_1_2_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E13:27_1_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:27_1_1_1_2_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:27_1_1_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="arytm_3.html#K2" target="_self">+^</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">l</font></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal3.html#T78">ORDINAL3:78</a></i>;<br/><a NAME="E5:27_1_1_1_2_1"/>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:27_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">l</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">k</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">l</font></span>)</span></span>)</span> &amp; <font color="Maroon">l</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">n</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">l</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>;<br/><a NAME="E7:27_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">l</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">l</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">l</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">l</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E8:27_1_1_1_2_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">k</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">l</font> &amp; <font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T3" target="_self">Th3</a>, <a class="txt" href="arytm_3.html#E10:27_1_1_1_2"><i><font color="Green">E17</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1_1_1_2_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:27_1_1_1_2_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">m</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E3:27_1_1_1_2_1_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">m</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E4:27_1_1_1_2_1_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E5:27_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">m</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="txt" href="arytm_3.html#E8:27_1_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E6:27_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">l</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">m</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T1" target="_self">Th1</a>, , </i>;<br/><a NAME="E7:27_1_1_1_2_1_1"/><b>then </b>
not <font color="Maroon">m</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/><b>hence </b><a NAME="E8:27_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><a NAME="E10:27_1_1_1_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">l</font></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T3" target="_self">Th3</a>, <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/><b>hence </b><a NAME="E11:27_1_1_1_2_1"/>
<font color="Maroon">l</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:27_1_1_1_2"/>
<font color="Maroon">l</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
