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





<b>assume </b><a NAME="E1:17"/><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Maroon">k</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<b>then consider </b><font color="Maroon">t1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:17"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">t2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:17"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E7:17"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">t2</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:17"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">t2</font></span>)</span>
 
;<br/>
<a NAME="E9:17"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font>, <font color="Maroon">m</font> = <font color="Maroon">m</font>, <font color="Maroon">k</font> = <font color="Maroon">k</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E11:17"/>
<font color="Maroon">k</font>,<font color="Maroon">m</font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i>, <a class="ref" href="euler_2.html#T1">EULER_2:1</a></i>;<br/>
<a NAME="E12:17"/><b>then </b>
<font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="int_2.html#T40">INT_2:40</a></i>;<br/>
<b>then consider </b><font color="Maroon">tt</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E14:17"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">tt</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E15:17"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">tt</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font>, <font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> = 
<font color="Maroon">n</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
1 <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="euler_1.html#T13">EULER_1:13</a></i>
<br/>.= 
1 <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="nat_1.html#T64">NAT_1:64</a></i>
;<br/>
<b>hence </b><a NAME="E18:17"/>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>


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