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<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">m</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">a</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:25"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">m</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="E3:25"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">mm</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E5:25"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">mm</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
;<br/>
<a NAME="E6:25"/>
 <a href="euler_1.html#K1">Euler</a> <font color="Maroon">m</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="euler_1.html#T21">EULER_1:21</a></i>;<br/>
<a NAME="E7:25"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">mm</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:25"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">mm</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">a</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">a</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:25"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</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">a</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>


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