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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c1">n</font> iff <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c2">m</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E11">A1</font></i>: 
( ( <font color="Maroon" title="c2">m</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c2">m</font></span>)</span> ) or ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp; not <font color="Maroon" title="c2">m</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c1">n</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c3">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="E3:11"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">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/>
<a NAME="E4:11"/>
( <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a></i>;<br/>
<a NAME="E5:11"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E1:11"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="arytm_3.html#E3:11"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<a NAME="E6:11"/><b>then </b>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:11"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/>
<b>hence </b><a NAME="E7:11"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:11"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="arytm_3.html#E3:11"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="arytm_3.html#D3" target="_self" title="ARYTM_3:def.3">Def3</a>, <a class="ref" href="ordinal3.html#D7" title="ORDINAL3:def.7">ORDINAL3:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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