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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="E1:23"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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/>

<a NAME="E2:23"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <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> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E3:23"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="arytm_3.html#K3" target="_self">*^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<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></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a>, <a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>;<br/>
<a NAME="E4:23"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E2:23"><i><font color="Green">E15</font></i></a>, <a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a></i>;<br/>
<a NAME="E5:23"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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> 1 implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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> 1
 ;<br/>

<a NAME="E2:23_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="ordinal3.html#T41">ORDINAL3:41</a></i>;<br/>
<b>hence </b><a NAME="E3:23_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:23"><i><font color="Green">E15</font></i></a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:23"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E3:23"><i><font color="Green">E16</font></i></a>, <a class="txt" href="arytm_3.html#E4:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="ordinal3.html#T36">ORDINAL3:36</a></i>;<br/>


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