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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <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="E1:3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">j</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:3"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:3"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">a'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font>
 ;<br/>
<b>take </b>
<font color="Maroon">a'</font>
;<br/>

<b>thus </b><a NAME="E7:3"/>
( <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">a'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i>, , , <a class="ref" href="euler_2.html#T10">EULER_2:10</a>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>


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