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<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_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="45_1_1"><b>suppose </b></a><a NAME="E1:45_1_1"/><i><font color="Green">E19</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">a</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> = 
0
<b>by </b><i><a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><b>suppose </b></a><a NAME="E1:45_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:45_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:45_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/><a NAME="E4:45_1_2"/>
<font color="Maroon">a</font> <a href="int_1.html#K6">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>, <a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/><b>hence </b><a NAME="E5:45_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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