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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>thus </b><a NAME="E1:36"/>
( <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> implies <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1"/>
<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>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green">E22</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> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:36_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 
<b>by </b><i><a class="txt" href="nat_1.html#E3:36_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:36_1_1"/><i><font color="Green">E25</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:36_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <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></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K4" target="_self">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:36_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="nat_1.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E4:36_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K4" target="_self">mod</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:36_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nat_1.html#E2:36_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nat_1.html#T46" target="_self">Th46</a></i>;<br/><b>hence </b><a NAME="E5:36_1_1"/>
<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>
 <b>by </b><i><a class="txt" href="nat_1.html#E4:36_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="nat_1.html#E1:36_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nat_1.html#E2:36_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_1.html#E3:36_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nat_1.html#T43" target="_self">Th43</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:36_1"/>
<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>
 <b>by </b><i><a class="txt" href="nat_1.html#E3:36_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:36"/>
<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>
 ;<br/>

<b>hence </b><a NAME="E3:36"/>
<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>
 <b>by </b><i><a class="ref" href="nat_1.html#D3" target="_self">Def3</a></i>;<br/>


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