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

<b>let </b><font color="Maroon">Gc</font> be  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Gc</font>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Gc</font> <a href="hidden.html#R1">=</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">g</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:14"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">Gc</font> is <a href="group_1.html#V6">finite</a> implies  ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i1</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i1</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">Gc</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">zero</font> = 0, <font color="Maroon">i0</font> =  <a href="group_1.html#K9">ord</a> <font color="Maroon">Gc</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E3:14_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">zero</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font>
 
<b>by </b><i>, <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">zero</font> = 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font>
<b>by </b><i><a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i0</font>
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T29">GR_CY_1:29</a></i>
;<br/>
<b>hence </b><a NAME="E5:14_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i1</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i1</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:14"/>
(  ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i1</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i1</font> ) implies <font color="Maroon">Gc</font> is <a href="group_1.html#V6">finite</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">i1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><a NAME="E1:14_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i1</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> or <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</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="14_2_1_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 ;<br/><div class="add"><a NAME="E2:14_2_1_1_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 0
 ;<br/><a NAME="E3:14_2_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>set </b><font color="Maroon">r</font> = <font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>;<br/><a NAME="E4:14_2_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="group_1.html#T70">GROUP_1:70</a></i>;<br/>
<a NAME="E2:14_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E3:14_2_1_1_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>;<br/>
<b>hence </b><a NAME="E4:14_2_1_1_1_1"/>
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E5:14_2_1_1_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">i2</font> being  <a href="int_1.html#NM1">Integer</a>  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i2</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<i><font color="Green">E67</font></i>: <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i2</font> = 
<font color="Maroon">g</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"><font color="Maroon">i2</font> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="newton.html#T80">NEWTON:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T63">GROUP_1:63</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">Gc</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#T67">GROUP_1:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T61">GROUP_1:61</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E2:14_2_1_1_1_2"/>
<font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E4:14_2_1_1_1_2"/>
( <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="newton.html#T79">NEWTON:79</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E7:14_2_1_1_1"/>
 ex <font color="Olive">F</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st  <a href="relat_1.html#K2">rng</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_3"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">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">Gc</font> = <font color="Maroon">g</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="E2:14_2_1_1_1_3"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E3:14_2_1_1_1_3"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">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> <font color="Maroon">m</font> 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="E4:14_2_1_1_1_3"/><i><font color="Green">E70</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> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:14_2_1_1_1_3"/><i><font color="Green">E71</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#M1">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">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_3_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="E1:14_2_1_1_1_3_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="E3:14_2_1_1_1_3_1"/><i><font color="Green">E72</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="E4:14_2_1_1_1_3_1"/><i><font color="Green">E73</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#M1">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="E6:14_2_1_1_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</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="E6:14_2_1_1_1_3"/><i><font color="Green">E82</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 <font color="Maroon">Gc</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_1_1_3_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 <font color="Maroon">Gc</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_3_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="E1:14_2_1_1_1_3_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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:14_2_1_1_1_3_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:14_2_1_1_1_3_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:14_2_1_1_1_3_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 <font color="Maroon">Gc</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:14_2_1_1_1_3"/><i><font color="Green">E84</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font> <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="14_2_1_1_1_3_3"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_1_1_3_3"/>
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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font> holds <br/><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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_3_3_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="E1:14_2_1_1_1_3_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Gc</font> ;<br/>
<a NAME="E3:14_2_1_1_1_3_3_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Gc</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:14_2_1_1_1_3_3_1"/><b>then </b><i><font color="Green">E85</font></i>: 
 ex <font color="Olive">i2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i2</font>
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>
<a NAME="E5:14_2_1_1_1_3_3_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_3_3_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:14_2_1_1_1_3_3_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:14_2_1_1_1_3_3_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>and </b><br/><a NAME="E4:14_2_1_1_1_3_3_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_3_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1_1_3_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1_1_1_3_3_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <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="14_2_1_1_1_3_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_1_1_3_3_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:14_2_1_1_1_3_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/><a NAME="E3:14_2_1_1_1_3_3_1_1_1_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_3_3_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_1_1_3_3_1_1_1_1_1_1"/>
<font color="Maroon">m</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="E2:14_2_1_1_1_3_3_1_1_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>


</div><b>end;</b></div><a NAME="E4:14_2_1_1_1_3_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:14_2_1_1_1_3_3_1_1_1_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">m</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#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:14_2_1_1_1_3_3_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:14_2_1_1_1_3_3_1_1_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1_1_3_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_1_1_3_3_1_1_1_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:14_2_1_1_1_3_3_1_1_1_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1_3_3_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_1_1_3_3_1_1_1_1_2_1"/>
<font color="Maroon">n</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="E2:14_2_1_1_1_3_3_1_1_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>


</div><b>end;</b></div><a NAME="E3:14_2_1_1_1_3_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:14_2_1_1_1_3_3_1_1_1_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">n</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#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:14_2_1_1_1_3_3_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:14_2_1_1_1_3_3_1_1_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:14_2_1_1_1_3_3_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:14_2_1_1_1_3_3_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>
 <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="E2:14_2_1_1_1_3_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font> <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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E8:14_2_1_1_1_3"/>
 <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 <font color="Maroon">Gc</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:14_2_1_1_1"/>
<font color="Maroon">Gc</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_1_2"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:14_2_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> 0
 ;<br/><a NAME="E3:14_2_1_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>set </b><font color="Maroon">r</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font>;<br/><a NAME="E4:14_2_1_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i1</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="group_1.html#T70">GROUP_1:70</a></i>;<br/>
<a NAME="E2:14_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E3:14_2_1_1_2_1"/><b>then </b>
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>;<br/>
<b>hence </b><a NAME="E4:14_2_1_1_2_1"/>
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E5:14_2_1_1_2"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">i2</font> being  <a href="int_1.html#NM1">Integer</a>  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i2</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<i><font color="Green">E104</font></i>: <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i2</font> = 
<font color="Maroon">g</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"><font color="Maroon">i2</font> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="newton.html#T80">NEWTON:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T63">GROUP_1:63</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">Gc</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#T67">GROUP_1:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">Gc</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T61">GROUP_1:61</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E2:14_2_1_1_2_2"/>
<font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">i2</font> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E4:14_2_1_1_2_2"/>
( <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font> )
 <b>by </b><i>, , <a class="ref" href="newton.html#T79">NEWTON:79</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E7:14_2_1_1_2"/>
 ex <font color="Olive">F</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st  <a href="relat_1.html#K2">rng</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">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">Gc</font> = <font color="Maroon">g</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="E2:14_2_1_1_2_3"/><i><font color="Green">E105</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E3:14_2_1_1_2_3"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">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> <font color="Maroon">m</font> 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="E4:14_2_1_1_2_3"/><i><font color="Green">E107</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> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:14_2_1_1_2_3"/><i><font color="Green">E108</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#M1">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">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_3_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="E1:14_2_1_1_2_3_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="E3:14_2_1_1_2_3_1"/><i><font color="Green">E109</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="E4:14_2_1_1_2_3_1"/><i><font color="Green">E110</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#M1">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="E6:14_2_1_1_2_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</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="E6:14_2_1_1_2_3"/><i><font color="Green">E111</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 <font color="Maroon">Gc</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_1_2_3_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 <font color="Maroon">Gc</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_3_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="E1:14_2_1_1_2_3_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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:14_2_1_1_2_3_2_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:14_2_1_1_2_3_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:14_2_1_1_2_3_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 <font color="Maroon">Gc</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:14_2_1_1_2_3"/><i><font color="Green">E113</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font> <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="14_2_1_1_2_3_3"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_1_2_3_3"/>
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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font> holds <br/><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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_3_3_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="E1:14_2_1_1_2_3_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Gc</font> ;<br/>
<a NAME="E3:14_2_1_1_2_3_3_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Gc</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:14_2_1_1_2_3_3_1"/><b>then </b><i><font color="Green">E114</font></i>: 
 ex <font color="Olive">i2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i2</font>
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>
<a NAME="E5:14_2_1_1_2_3_3_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_3_3_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:14_2_1_1_2_3_3_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:14_2_1_1_2_3_3_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>and </b><br/><a NAME="E4:14_2_1_1_2_3_3_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_3_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1_2_3_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1_1_2_3_3_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <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="14_2_1_1_2_3_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_1_2_3_3_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:14_2_1_1_2_3_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/><a NAME="E3:14_2_1_1_2_3_3_1_1_1_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_3_3_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_1_2_3_3_1_1_1_1_1_1"/>
<font color="Maroon">m</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="E2:14_2_1_1_2_3_3_1_1_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>


</div><b>end;</b></div><a NAME="E4:14_2_1_1_2_3_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:14_2_1_1_2_3_3_1_1_1_1_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">m</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#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:14_2_1_1_2_3_3_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:14_2_1_1_2_3_3_1_1_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1_2_3_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_1_2_3_3_1_1_1_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:14_2_1_1_2_3_3_1_1_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_3_3_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_1_2_3_3_1_1_1_1_2_1"/>
<font color="Maroon">n</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="E2:14_2_1_1_2_3_3_1_1_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>


</div><b>end;</b></div><a NAME="E3:14_2_1_1_2_3_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:14_2_1_1_2_3_3_1_1_1_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">n</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#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:14_2_1_1_2_3_3_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:14_2_1_1_2_3_3_1_1_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:14_2_1_1_2_3_3_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</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">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:14_2_1_1_2_3_3_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>
 <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="E2:14_2_1_1_2_3_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font> <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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E8:14_2_1_1_2_3"/>
 <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 <font color="Maroon">Gc</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:14_2_1_1_2"/>
<font color="Maroon">Gc</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_2"/>
<font color="Maroon">Gc</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
( <font color="Maroon">Gc</font> is <a href="group_1.html#V6">finite</a> iff  ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i1</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i1</font> ) )
 <b>by </b><i/>;<br/>


</div>
