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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c1">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">H</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><a NAME="E2:27"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c3">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<a NAME="E3:27"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c3">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E2:27"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="group_2.html#T75" title="GROUP_2:th.75">GROUP_2:75</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font>;<br/>
<a NAME="E5:27"/>
<font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T50" title="GROUP_2:th.50">GROUP_2:50</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font><b> such that </b><br/><a NAME="E8:27"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="group_8.html#E3:27"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="group_8.html#T23" target="_self" title="GROUP_8:th.23">Th23</a></i>;<br/>
<a NAME="E9:27"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:27"/><b>then </b>
<font color="Maroon" title="c6">c</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T49" title="GROUP_2:th.49">GROUP_2:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c6">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E12:27"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:27"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="group_8.html#E9:27"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="group_2.html#T49" title="GROUP_2:th.49">GROUP_2:49</a></i>;<br/>
<a NAME="E13:27"/>
 ex <font color="Olive" title="b1">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/>
for <font color="Olive" title="b1">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  holds <br/>(  not <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">j</font> or  not <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:27_1"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
;<br/>
<b>consider </b><font color="Maroon" title="c8">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:27_1"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="group_8.html#E12:27"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="gr_cy_1.html#T25" title="GR_CY_1:th.25">GR_CY_1:25</a></i>;<br/>
<a NAME="E5:27_1"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="group_8.html#E2:27_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="group_8.html#E4:27_1"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/>
<a NAME="E6:27_1"/>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E8:27"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="group_8.html#E4:27_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="group_2.html#T53" title="GROUP_2:th.53">GROUP_2:53</a></i>;<br/>
<b>hence </b><a NAME="E7:27_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E5:27_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="group_1.html#T59" title="GROUP_1:th.59">GROUP_1:59</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c8">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E15:27"/><i><font color="Green" title="E24">A10</font></i>: 
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E17:27"/><i><font color="Green" title="E25">A11</font></i>: 
( <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> or <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c9">n</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8" title="INT_1:th.8">INT_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2"/>
( <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> or <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c9">n</font> )
 </span><b>by </b><i><a class="txt" href="group_8.html#E17:27"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1"><b>suppose </b></a><a NAME="E1:27_2_1"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font>
 ;<br/><div class="add"><a NAME="E2:27_2_1"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="group_8.html#E15:27"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="group_8.html#E1:27_2_1"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/><a NAME="E3:27_2_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="group_8.html#E15:27"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="group_8.html#E1:27_2_1"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E4:27_2_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">k</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">H</font> )
 <b>by </b><i><a class="txt" href="group_8.html#E9:27"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="group_8.html#E2:27_2_1"><i><font color="Green" title="E27">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_2"><b>suppose </b></a><a NAME="E1:27_2_2"/><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c9">n</font>
 ;<br/><div class="add"><a NAME="E2:27_2_2"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="group_8.html#E15:27"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="group_8.html#E1:27_2_2"><i><font color="Green" title="E26">A14</font></i></a></i>;<br/><a NAME="E3:27_2_2"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font>
 ;<br/><a NAME="E4:27_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="group_8.html#E9:27"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="group_8.html#E15:27"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="group_8.html#E1:27_2_2"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="group_1.html#T71" title="GROUP_1:th.71">GROUP_1:71</a></i>;<br/><a NAME="E5:27_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></i>;<br/><a NAME="E6:27_2_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>;<br/><b>hence </b><a NAME="E7:27_2_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">k</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">H</font> )
 <b>by </b><i><a class="txt" href="group_8.html#E3:27_2_2"><i><font color="Green" title="E27">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
