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<b>let </b><font color="Maroon">a</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">b</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:17"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E2:17"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">c</font>, <font color="Olive">d1</font>, <font color="Olive">d2</font> being <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>  holds <br/>(  not <font color="Olive">d1</font>,<font color="Olive">d2</font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  or  not <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Olive">d1</font> or  not <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Olive">d2</font> )
 ;<br/>

<a NAME="E3:17"/><i><font color="Green">E15</font></i>: 
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1"/>
( <font color="Maroon">a</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">b</font> or <font color="Maroon">b</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">a</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">A</font> = <font color="Maroon">b</font>;<br/><b>take </b><font color="Maroon">B</font> = <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E2:17_1_1_1"/>
( <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="txt" href="arytm_3.html#E1:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>thus </b><a NAME="E3:17_1_1_1"/>
for <font color="Olive">c</font>, <font color="Olive">d1</font>, <font color="Olive">d2</font> being <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>  holds <br/>(  not <font color="Olive">d1</font>,<font color="Olive">d2</font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  or  not <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Olive">d1</font> or  not <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Olive">d2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">b</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">A</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">B</font> = <font color="Maroon">b</font>;<br/><b>thus </b><a NAME="E2:17_1_1_2"/>
( <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="txt" href="arytm_3.html#E1:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>thus </b><a NAME="E3:17_1_1_2"/>
for <font color="Olive">c</font>, <font color="Olive">d1</font>, <font color="Olive">d2</font> being <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>  holds <br/>(  not <font color="Olive">d1</font>,<font color="Olive">d2</font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  or  not <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Olive">d1</font> or  not <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Olive">d2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:17"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>and </b><br/><a NAME="E6:17"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C</font>] holds <br/><font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:17"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">c</font>, <font color="Olive">d1</font>, <font color="Olive">d2</font> being <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>  holds <br/>(  not <font color="Olive">d1</font>,<font color="Olive">d2</font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  or  not <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Olive">d1</font> or  not <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Olive">d2</font> ) ) )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:17"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font>, <font color="Maroon">B</font> = <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  <b>by </b><i><a class="txt" href="arytm_3.html#E3:17"><i><font color="Green">E15</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E10:17"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 1 <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>
<a NAME="E11:17"/><b>then </b>
not <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:17"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">d1</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">d2</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E13:17"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">d2</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <b>by </b><i/>;<br/>
<a NAME="E14:17"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:17"><i><font color="Green">E15</font></i></a>, , <a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a>, <a class="ref" href="ordinal2.html#T55">ORDINAL2:55</a></i>;<br/>
<a NAME="E15:17"/><b>then </b>
( <font color="Maroon">c</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="ordinal3.html#T39">ORDINAL3:39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font>, <font color="Maroon">d1</font> = <font color="Maroon">d1</font>, <font color="Maroon">d2</font> = <font color="Maroon">d2</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#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E17:17"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E18:17"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">d2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">d1</font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:17"><i><font color="Green">E15</font></i></a>, , <a class="ref" href="ordinal3.html#T38">ORDINAL3:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c'</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">d1'</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">d2'</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">d1'</font>,<font color="Maroon">d2'</font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  &amp; <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">d1'</font> &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">d2'</font> )
 ;<br/><a NAME="E2:17_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c'</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">d1'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c'</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">d2'</font> )
 <b>by </b><i>, <a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>;<br/><b>hence </b><a NAME="E3:17_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E3:17"><i><font color="Green">E15</font></i></a>, <a class="txt" href="arytm_3.html#E6:17"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:17"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">d1</font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E2:17"><i><font color="Green">E14</font></i></a>, , <a class="txt" href="arytm_3.html#E5:17"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E21:17"/>
( 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> or <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> 1 )
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/>
<a NAME="E22:17"/><b>then </b>
1 <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="ordinal3.html#T17">ORDINAL3:17</a>, <a class="ref" href="ordinal3.html#T22">ORDINAL3:22</a></i>;<br/>
<a NAME="E23:17"/><b>then </b>
<font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>
<b>hence </b><a NAME="E24:17"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E6:17"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>


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