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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="c1">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">m</font> &amp; <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> implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )</span><br/>



<b>assume </b><a NAME="E1:12"/>
( <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> &amp; <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"> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font></span><br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green" title="E12">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="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <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> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T11" target="_self" title="ARYTM_3:th.11">Th11</a></i>;<br/>
<a NAME="E3:12"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<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>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a>, <a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></i>;<br/>
<a NAME="E4:12"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 
<b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a></i>;<br/>
<a NAME="E5:12"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">n</font></span>)</span> <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> 1 implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">n</font></span>)</span> <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> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font></span><br/>

<a NAME="E2:12_1"/><b>then </b>
<font color="Maroon" title="c2">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="ordinal3.html#T41" title="ORDINAL3:th.41">ORDINAL3:41</a></i>;<br/>
<b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:12"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="#E3:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="#E4:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="ordinal3.html#T36" title="ORDINAL3:th.36">ORDINAL3:36</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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