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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">l</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>





<b>assume </b><a NAME="E1:26"/>
<font color="Maroon" title="c1">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">a</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:26"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T9" target="_self" title="ARYTM_3:th.9">Th9</a></i>;<br/>
<b>assume </b><a NAME="E4:26"/>
<font color="Maroon" title="c1">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">+^</a> <font color="Maroon" title="c3">l</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">b</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E6:26"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">+^</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T9" target="_self" title="ARYTM_3:th.9">Th9</a></i>;<br/>
<b>assume </b><a NAME="E7:26"/><i><font color="Green" title="E21">A3</font></i>: 
not <font color="Maroon" title="c1">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c3">l</font>
 ;<br/>

<font color="Maroon" title="c3">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_3.html#E3:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="arytm_3.html#E6:26"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="ordinal3.html#T65" title="ORDINAL3:th.65">ORDINAL3:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="ordinal3.html#T76" title="ORDINAL3:th.76">ORDINAL3:76</a></i>
;<br/>
<b>hence </b><a NAME="E9:26"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E7:26"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="arytm_3.html#D3" target="_self" title="ARYTM_3:def.3">Def3</a></i>;<br/>


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