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

<b>let </b><font color="Maroon">n</font> be   <a href="int_1.html#NM1">Integer</a>, <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="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_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="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1"/>
( ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies <font color="Maroon">a</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) &amp; ( 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font> implies <font color="Maroon">a</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:42_1_2_1"/><i><font color="Green">E22</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E3:42_1_2_1"/><i><font color="Green">E23</font></i>: 
<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#E5"><i><font color="Green">Lemma29</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">nn</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></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="E6:42_1_2_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">aa</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">nn</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">aa</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">nn</font></span>)</span> &amp; <font color="Maroon">aa</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">nn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">nn</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/><a NAME="E7:42_1_2_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:42_1_2_1_1"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E3:42_1_2_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<a NAME="E4:42_1_2_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E6"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:42_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">nn</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">aa</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">nn</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">nn</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E6:42_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_3.html#E5"><i><font color="Green">Lemma29</font></i></a>, , <a class="txt" href="int_3.html#E7"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:42_1_2_1"/>
<font color="Maroon">a</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E3:42_1_2"/><i><font color="Green">E44</font></i>: 
( 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E4:42_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><a NAME="E5:42_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E6:42_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <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="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/><a NAME="E7:42_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="binop_2.html#K7">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_2_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="int_3.html#E6"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E2:42_1_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="int_3.html#E5"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E3:42_1_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</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">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1_2_2"/>
 <a href="binop_2.html#K7">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:42_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="binop_2.html#K7">-</a> 1
 ;<br/>

<a NAME="E2:42_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:42_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E4:42_1_2_3"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E5:42_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E6"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E6:42_1_2_3"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E7:42_1_2_3"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E8:42_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="int_3.html#E5"><i><font color="Green">Lemma29</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:42_1_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1
 <b>by </b><i>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E10:42_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>;<br/><b>hence </b><a NAME="E11:42_1_2"/>
<font color="Maroon">a</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
