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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="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:51"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E3:51"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="pepin.html#E1:51"><i><font color="Green">E40</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E4:51"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:51_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/>
<a NAME="E3:51_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E1:51"><i><font color="Green">E40</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:51_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E1:51"><i><font color="Green">E40</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:51"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="pepin.html#E1:51"><i><font color="Green">E40</font></i></a>, <a class="ref" href="euler_2.html#T34">EULER_2:34</a></i>;<br/>
<a NAME="E6:51"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="pepin.html#E3:51"><i><font color="Green">E42</font></i></a>, <a class="txt" href="pepin.html#E4:51"><i><font color="Green">E43</font></i></a>, <a class="ref" href="pepin.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E7:51"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="pepin.html#E1:51"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E8:51"/>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="pepin.html#E4:51"><i><font color="Green">E43</font></i></a>, <a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<b>hence </b><a NAME="E9:51"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E2:51"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pepin.html#E4:51"><i><font color="Green">E43</font></i></a>, <a class="txt" href="pepin.html#E6:51"><i><font color="Green">E44</font></i></a>, <a class="txt" href="pepin.html#E7:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euler_2.html#T27">EULER_2:27</a></i>;<br/>


</div>
