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<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:72"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:72"/>
 ex <font color="Olive">p</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">k</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="Olive">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<a NAME="E1:72_1"/>
<font color="Maroon">k</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> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 
;<br/>
<b>hence </b><a NAME="E2:72_1"/>
 ex <font color="Olive">p</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">k</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="Olive">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:72"/>
<font color="Maroon">k</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>


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