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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#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:38"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:38"/><b>then </b><i><font color="Green">E23</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="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K3" target="_self">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="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K3" target="_self">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E3:38"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K3" target="_self">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K3" target="_self">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E4:38"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 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="nat_1.html#E2:38"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E5:38"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K3" target="_self">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K3" target="_self">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="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K3" target="_self">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K3" target="_self">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:38_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K3" target="_self">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="nat_1.html#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E3:38_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="nat_1.html#E2:38"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:38"/>
<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="nat_1.html#E3:38"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nat_1.html#E4:38"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>


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