<?xml version="1.0"?>
<div class="add">

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





<b>assume </b><a NAME="E1:61"/><i><font color="Green">E31</font></i>: 
<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> 0
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><b>suppose </b></a><a NAME="E1:61_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font> = 
0
<b>by </b><i/>
<br/>.= 
<font color="Maroon">l</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2"><b>suppose </b></a><a NAME="E1:61_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:61_1_2"/><i><font color="Green">E35</font></i>: 
( <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="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:61_1_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">l</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="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">l</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:61_1_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l</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> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:61_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nat_1.html#E3:61_1_2"><i><font color="Green">E35</font></i></a></i>;<br/><b>hence </b><a NAME="E7:61_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E3:61_1_2"><i><font color="Green">E35</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
