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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> ;<br/><b>let </b><font color="Maroon">p</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="E1:15"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> )
 ;<br/>

<b>set </b><font color="Maroon">T</font> = <font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:15"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E4:15"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then </b><font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<b>hence </b><a NAME="E6:15"/>
<font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font>
 ;<br/>


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