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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>





<b>assume </b><a NAME="E1:71"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:71"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="ordinal2.html#T55">ORDINAL2:55</a></i>;<br/>
<a NAME="E3:71"/><b>then </b><i><font color="Green">E45</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K6" target="_self">div^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal3.html#D7" target="_self">Def7</a></i>;<br/>
<b>assume </b><a NAME="E4:71"/><i><font color="Green">E46</font></i>: 
(  not <font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K6" target="_self">div^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K7" target="_self">mod^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E5:71"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K6" target="_self">div^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K7" target="_self">mod^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T78" target="_self">Th78</a></i>;<br/>
<a NAME="E6:71"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K6" target="_self">div^</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal3.html#E3:71"><i><font color="Green">E45</font></i></a>, <a class="txt" href="ordinal3.html#E4:71"><i><font color="Green">E46</font></i></a>, <a class="ref" href="ordinal3.html#T24" target="_self">Th24</a>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>
<a NAME="E7:71"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K6" target="_self">div^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal3.html#K6" target="_self">div^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E3:71"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ordinal3.html#T27" target="_self">Th27</a>, <a class="ref" href="ordinal2.html#T58">ORDINAL2:58</a></i>;<br/>
<b>hence </b><a NAME="E8:71"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:71"><i><font color="Green">E44</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>


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