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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>

<a NAME="E2:54"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">m</font>,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E4:54_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E1:54"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:54_1"/>
<font color="Maroon">m</font>,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="int_1.html#D3">INT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:54"/>
( <font color="Maroon">m</font>,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">n</font> implies <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_2"/>
<font color="Maroon">m</font>,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">n</font>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:54_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#D3">INT_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font>, <font color="Maroon">n</font> = <font color="Maroon">n</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<i><font color="Green">E51</font></i>: <font color="Maroon">m</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="pepin.html#E1:54"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
1 <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="euler_1.html#T13">EULER_1:13</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font>, <font color="Maroon">n</font> = <font color="Maroon">n</font>, <font color="Maroon">1'</font> = 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> = 
<font color="Maroon">m</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<font color="Maroon">1'</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="pepin.html#E2:54"><i><font color="Green">E45</font></i></a>, <a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<font color="Maroon">1'</font>
<b>by </b><i>, <a class="ref" href="nat_1.html#T64">NAT_1:64</a></i>
;<br/>
<b>hence </b><a NAME="E8:54_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:54"/>
( <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 iff <font color="Maroon">m</font>,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="pepin.html#E1"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


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