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

<b>let </b><font color="Maroon" title="c1">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">a</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c1">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Olive" title="b1">a</font> ) )</span><br/>

<b>assume </b><a NAME="E1:87"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c1">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Olive" title="b1">a</font> )</span><br/>

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c1">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<div><i><font color="Green" title="E40">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1"/>
<font color="Maroon" title="c2">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font></span><br/><a NAME="E2:87_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="int_1.html#D8" target="_self" title="INT_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E3:87_1"/>
<font color="Maroon" title="c1">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" target="_self" title="INT_1:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_2"/>
<font color="Maroon" title="c1">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>then consider </b><font color="Maroon" title="c3">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:87_2"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" target="_self" title="INT_1:def.9">Def9</a></i>;<br/><i><font color="Green" title="E42">A4</font></i>: <font color="Maroon" title="c2">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> = 
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="int_1.html#D8" target="_self" title="INT_1:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span>

;<br/><a NAME="E5:87_2"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:87_2_2"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:87_2_2"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E3:87_2_2"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E4:87_2_2"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
;<br/>
<a NAME="E5:87_2_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1
 
<b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:87_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:87_2"><i><font color="Green" title="E41">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E6:87_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:87_2_3"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>
<a NAME="E2:87_2_3"/><b>then </b><i><font color="Green" title="E44">A7</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:87_2_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
 
<b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>
<a NAME="E4:87_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:87_2"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E2:87_2_3"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E5:87_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
 
;<br/>
<a NAME="E6:87_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E7:87_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/>
<a NAME="E8:87_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 
;<br/>
<a NAME="E9:87_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:87_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E7:87_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="#E5:87_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="int_1.html#D4" target="_self" title="INT_1:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E8:87_2"/>
<font color="Maroon" title="c2">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E3:87_2"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E4:87_2"><i><font color="Green" title="E42">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:87"/>
( <font color="Maroon" title="c2">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c1">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="#E2:87"><i><font color="Green" title="E40">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
