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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green">E14</font></i>: 
( ( <font color="Maroon">c<sub>2</sub></font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>1</sub></font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:22"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E4:22"/>
( <a href="xboole_0.html#K1">{}</a>  <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <a href="xboole_0.html#K1">{}</a>  <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a></i>;<br/>
<a NAME="E5:22"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E1:22"><i><font color="Green">E14</font></i></a>, <a class="txt" href="arytm_3.html#E3:22"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E6:22"/><b>then </b>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="arytm_3.html#K2" target="_self">+^</a> <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:22"><i><font color="Green">E15</font></i></a>, <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a>, <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<b>hence </b><a NAME="E7:22"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:22"><i><font color="Green">E14</font></i></a>, <a class="txt" href="arytm_3.html#E3:22"><i><font color="Green">E15</font></i></a>, <a class="ref" href="arytm_3.html#D3" target="_self">Def3</a>, <a class="ref" href="ordinal3.html#D7">ORDINAL3:def 7</a></i>;<br/>


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