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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<a NAME="E2:13"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i><a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> = <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:13"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:13"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">p</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E6:13"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:13"/>
<font color="Maroon">a</font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:13"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/>
<a NAME="E9:13"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E10:13"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">x'</font> = 1;<br/>
<a NAME="E1:13_1_1"/><i><font color="Green">E65</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E2:13_1_1_1"/><b>then </b>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> 1
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_1.html#T44">GROUP_1:44</a></i>
;<br/>
<b>hence </b><a NAME="E3:13_1_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:13"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:13_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E6:13_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:13_2"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_2_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:13_2_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:13_2_2_1"/>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:13_2_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:13_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_3"/>
 ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">car</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<a NAME="E2:13_2_3_1"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:13_2_3_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E4:13_2_3_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">car</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:13_2_3_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">car</font>,<font color="Maroon">car</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E6:13_2_3_1"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> <font color="Maroon">car</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">car</font>,<font color="Maroon">car</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E7:13_2_3_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> <font color="Maroon">car</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">car</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_3_1_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> <font color="Maroon">car</font></span>)</span> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">car</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">f</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> <font color="Maroon">car</font>;<br/>
<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_2_3_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> <font color="Maroon">car</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_2_3_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> <font color="Maroon">car</font></span>)</span>
 <b>and </b><br/><a NAME="E4:13_2_3_1_1_1"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> <font color="Maroon">car</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">xp</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">yp</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:13_2_3_1_1_1"/><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xp</font>,<font color="Maroon">yp</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E7:13_2_3_1_1_1"/>
<font color="Maroon">xp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">car</font>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:13_2_3_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">xp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:13_2_3_1_1_1"/>
<font color="Maroon">yp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">car</font>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:13_2_3_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">yp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>
<i><font color="Green">E90</font></i>: <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48">GROUP_1:48</a></i>
;<br/>
<a NAME="E14:13_2_3_1_1_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">car</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_3_1_1_1_2"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>;<br/>
<a NAME="E1:13_2_3_1_1_1_2_1"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_3_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_2_3_1_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_3_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:13_2_3_1_1_1_2_1_1_1"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:13_2_3_1_1_1_2_1_1_1"/><i><font color="Green">E93</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><i><font color="Green">E94</font></i>: <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> = 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48">GROUP_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T82">GROUP_1:82</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42">GROUP_1:42</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> 0
<b>by </b><i>, <a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T82">GROUP_1:82</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><b>take </b><font color="Maroon">x</font> =  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E4:13_2_3_1_1_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_3_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:13_2_3_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:13_2_3_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:13_2_3_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>take </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>;<br/><a NAME="E4:13_2_3_1_1_1_2_1_1_2"/><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> = 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48">GROUP_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T82">GROUP_1:82</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42">GROUP_1:42</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E6:13_2_3_1_1_1_2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_2_3_1_1_1_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">car</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:13_2_3_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">car</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_2_3_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> <font color="Maroon">car</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">car</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">op</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> <font color="Maroon">car</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">car</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #);<br/>
<a NAME="E9:13_2_3_1"/>
 <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) is   <a href="group_1.html#NM1">Group</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_3_1_2"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #)<br/> for <font color="Olive">f'</font>, <font color="Olive">g'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f'</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">g'</font> holds <br/><font color="Olive">f'</font> <a href="group_1.html#K1">*</a> <font color="Olive">g'</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #), <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #);<br/><b>let </b><font color="Maroon">f'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">g'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:13_2_3_1_2_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> )
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">car</font>,<font color="Maroon">car</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E2:13_2_3_1_2_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="realset1.html#K1">||</a> <font color="Maroon">car</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span>
 
;<br/>
<b>thus </b><a NAME="E3:13_2_3_1_2_1"/>
<font color="Maroon">f'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:13_2_3_1_2"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #), <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #), <font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #);<br/>

<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:13_2_3_1_2_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">f'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:13_2_3_1_2_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h'</font>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f'</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E7:13_2_3_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:13_2_3_1_2"/>
 ex <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) st <br/>for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Olive">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_3_1_2_3"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">car</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_3_1_2_3_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">x'</font> =  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>;<br/>
<a NAME="E1:13_2_3_1_2_3_1_1"/><i><font color="Green">E110</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E2:13_2_3_1_2_3_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:13_2_3_1_2_3_1_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_2_3_1_2_3_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">car</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">e</font> =  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) <b>by </b><i><a class="ref" href="group_1.html#T82">GROUP_1:82</a></i>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<a NAME="E3:13_2_3_1_2_3"/>
for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #);<br/>

<b>reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:13_2_3_1_2_3_2"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_1"><b>proof </b></a><div class="add">

<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> = 
<font color="Maroon">h'</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h'</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E2:13_2_3_1_2_3_2_1"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:13_2_3_1_2_3_2"/>
<font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_2"><b>proof </b></a><div class="add">

<font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h'</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E2:13_2_3_1_2_3_2_2"/>
<font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:13_2_3_1_2_3_2"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) st <br/>( <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_3_1_2_3_2_3"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:13_2_3_1_2_3_2_3_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E3:13_2_3_1_2_3_2_3_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E5:13_2_3_1_2_3_2_3_1"/>
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:13_2_3_1_2_3_2_3"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:13_2_3_1_2_3_2_3"/><i><font color="Green">E114</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/>
<a NAME="E5:13_2_3_1_2_3_2_3"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">car</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_3_1_2_3_2_3_2"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_2_3_1_2_3_2_3_2_1_1"/>
(  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:13_2_3_1_2_3_2_3_2_1_1_1"/><i><font color="Green">E115</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">x</font> =  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>;<br/><a NAME="E2:13_2_3_1_2_3_2_3_2_1_1_1"/><i><font color="Green">E116</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><b>then </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> = 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="group_1.html#T82">GROUP_1:82</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>
;<br/><b>hence </b><a NAME="E4:13_2_3_1_2_3_2_3_2_1_1_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:13_2_3_1_2_3_2_3_2_1_1_2"/><i><font color="Green">E117</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <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#T18">INT_1:18</a></i>;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><a NAME="E3:13_2_3_1_2_3_2_3_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">r1</font> =  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>, <font color="Maroon">r2</font> = <font color="Maroon">n</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E2:13_2_3_1_2_3_2_3_2_1_1_2_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_2_1_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_3_1_2_3_2_3_2_1_1_2_1_1"/>
<font color="Maroon">r1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E2:13_2_3_1_2_3_2_3_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_3_1_2_3_2_3_2_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:13_2_3_1_2_3_2_3_2_1_1_2_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_2_1_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_3_1_2_3_2_3_2_1_1_2_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E2:13_2_3_1_2_3_2_3_2_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:13_2_3_1_2_3_2_3_2_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:13_2_3_1_2_3_2_3_2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_2_3_1_2_3_2_3_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">car</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) ;<br/>
<a NAME="E7:13_2_3_1_2_3_2_3"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>
 
;<br/>
<a NAME="E8:13_2_3_1_2_3_2_3"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_3"><b>proof </b></a><div class="add">

<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#T64">GROUP_1:64</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="group_1.html#T82">GROUP_1:82</a></i>
;<br/>
<b>hence </b><a NAME="E2:13_2_3_1_2_3_2_3_3"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:13_2_3_1_2_3_2_3"/>
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3_1_2_3_2_3_4"><b>proof </b></a><div class="add">

<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#T65">GROUP_1:65</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="group_1.html#T82">GROUP_1:82</a></i>
;<br/>
<b>hence </b><a NAME="E2:13_2_3_1_2_3_2_3_4"/>
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:13_2_3_1_2_3_2_3"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) st <br/>( <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:13_2_3_1_2_3_2"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:13_2_3_1_2_3"/>
for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:13_2_3_1_2"/>
 <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) is   <a href="group_1.html#NM1">Group</a>
 <b>by </b><i>, <a class="ref" href="group_1.html#T5">GROUP_1:5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">H1</font> =  <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">car</font>,<font color="Maroon">op</font> #) as   <a href="group_1.html#NM1">Group</a> ;<br/>
<a NAME="E11:13_2_3_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E12:13_2_3_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:13_2_3_1"/><b>then </b>
<font color="Maroon">H1</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E14:13_2_3_1"/>
 ex <font color="Olive">H</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">H</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:13_2_3"/><i><font color="Green">E124</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>
<a NAME="E4:13_2_3"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E5:13_2_3"/><b>then </b>
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_4.html#D5">GROUP_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:13_2_3"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:13_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:13"/>
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13_3"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:13_3"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E3:13_3"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E4:13_3"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">s</font>, <font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">z</font> =  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E8:13_3"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:13_3"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_3_1"/>
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> or <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_1_1"><b>suppose </b></a><a NAME="E1:13_3_1_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:13_3_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">s</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">p</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#T18">INT_1:18</a></i>;<br/><a NAME="E4:13_3_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i>, , , <a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/><a NAME="E5:13_3_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/><a NAME="E6:13_3_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T71">GROUP_1:71</a></i>;<br/><a NAME="E7:13_3_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T64">GROUP_1:64</a></i>;<br/><a NAME="E8:13_3_1_1"/><b>then </b><i><font color="Green">E136</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/><a NAME="E9:13_3_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E10:13_3_1_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:13_3_1_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_3_1_1_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">s'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:13_3_1_1_1"/>
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:13_3_1_1_1"/><b>then </b>
<font color="Maroon">s'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:13_3_1_1_1"/><b>then </b>
<font color="Maroon">s'</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">p'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">p'</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E5:13_3_1_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E12:13_3_1_1"/>
<font color="Maroon">a</font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E13:13_3_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_1_2"><b>suppose </b></a><a NAME="E1:13_3_1_2"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:13_3_1_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">p</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">s</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#T18">INT_1:18</a></i>;<br/><a NAME="E4:13_3_1_2"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i>, , , <a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/><a NAME="E5:13_3_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/><a NAME="E6:13_3_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T71">GROUP_1:71</a></i>;<br/><a NAME="E7:13_3_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T64">GROUP_1:64</a></i>;<br/><a NAME="E8:13_3_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/><a NAME="E9:13_3_1_2"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E10:13_3_1_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:13_3_1_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_3_1_2_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:13_3_1_2_1"/>
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">s'</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:13_3_1_2_1"/><b>then </b>
<font color="Maroon">p'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">s'</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:13_3_1_2_1"/><b>then </b>
<font color="Maroon">p'</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">s'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">s'</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">s'</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E5:13_3_1_2_1"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E12:13_3_1_2"/>
<font color="Maroon">a</font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E13:13_3_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:13"/><b>then </b><i><font color="Green">E148</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i>, , <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">ca</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:13"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">ca</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp;  <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">ca</font> )
 <b>by </b><i>, <a class="ref" href="group_1.html#D15">GROUP_1:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">So</font> =  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E17:13"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">So</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">ca</font>
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<b>hence </b><a NAME="E18:13"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>


</div>
