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

<b>let </b><font color="Maroon" title="c1">Gc</font> be  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c1">Gc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:14"/><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c1">Gc</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i1</font> &amp; <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">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" title="E15">A3</font></i>: 
<font color="Maroon" title="c1">Gc</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">zero</font> = 0, <font color="Maroon" title="c4">i0</font> =  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">Gc</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E3:14_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c3">zero</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i0</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:14_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">zero</font> = 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font>
<b>by </b><i><a class="ref" href="group_1.html#T43" title="GROUP_1:th.43">GROUP_1:43</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c4">i0</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:14_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="gr_cy_1.html#T29" title="GR_CY_1:th.29">GR_CY_1:29</a></i>
;<br/>
<b>hence </b><a NAME="E5:14_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i1</font> &amp; <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">i1</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_1"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:14"/>
(  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i1</font> &amp; <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">i1</font> ) implies <font color="Maroon" title="c1">Gc</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">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" title="c3">i</font>, <font color="Maroon" title="c4">i1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><a NAME="E1:14_2"/><i><font color="Green" title="E15">A5</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i1</font> &amp; <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c4">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" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i1</font> or <font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> )
 </span><b>by </b><i><a class="txt" href="gr_cy_2.html#E1:14_2"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i1</font>
 ;<br/><div class="add"><a NAME="E2:14_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 ;<br/><a NAME="E3:14_2_1_1_1"/><b>then </b><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><b>set </b><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font>;<br/><a NAME="E4:14_2_1_1_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">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" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c4">i1</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:14_2"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="group_1.html#T70" title="GROUP_1:th.70">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" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c4">i1</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E3:14_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>;<br/>
<b>hence </b><a NAME="E4:14_2_1_1_1_1"/>
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E5:14_2_1_1_1"/><i><font color="Green" title="E18">A8</font></i>: 
for <font color="Olive" title="b1">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">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" title="c6">i2</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<i><font color="Green" title="E19">A9</font></i>: <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">i2</font> = 
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="newton.html#T80" title="NEWTON:th.80">NEWTON:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="group_1.html#T67" title="GROUP_1:th.67">GROUP_1:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T61" title="GROUP_1:th.61">GROUP_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E2:14_2_1_1_1_2"/>
<font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="newton.html#T78" title="NEWTON:th.78">NEWTON:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">m</font>
;<br/>

<b>thus </b><a NAME="E4:14_2_1_1_1_2"/>
( <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="gr_cy_2.html#E1:14_2_1_1_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="newton.html#T79" title="NEWTON:th.79">NEWTON:79</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E7:14_2_1_1_1"/>
 ex <font color="Olive" title="b1">F</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> = <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c7">F</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E2:14_2_1_1_1_3"/><i><font color="Green" title="E19">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 <b>and </b><br/><a NAME="E3:14_2_1_1_1_3"/><i><font color="Green" title="E20">A11</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> holds <br/><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">p</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E4:14_2_1_1_1_3"/><i><font color="Green" title="E21">A12</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:14_2_1_1_1_3"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:14_2_1_1_1_3"/><i><font color="Green" title="E22">A13</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">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" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_2_1_1_1_3_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_2_1_1_1_3_1"/><i><font color="Green" title="E23">A14</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>and </b><br/><a NAME="E4:14_2_1_1_1_3_1"/><i><font color="Green" title="E24">A15</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1_3_1"><i><font color="Green" title="E23">A14</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">n</font>
;<br/>

<b>thus </b><a NAME="E6:14_2_1_1_1_3_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1_3"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1_3"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1_3_1"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1_3_1"><i><font color="Green" title="E24">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:14_2_1_1_1_3"/><i><font color="Green" title="E23">A16</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_2_1_1_1_3_2_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:14_2_1_1_1_3_2_1"/><i><font color="Green" title="E24">A17</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:14_2_1_1_1_3"><i><font color="Green" title="E22">A13</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:14_2_1_1_1_3_2_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1_3_2_1"><i><font color="Green" title="E24">A17</font></i></a></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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:14_2_1_1_1_3"/><i><font color="Green" title="E24">A18</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">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" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">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" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_2_1_1_1_3_3_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font> ;<br/>
<a NAME="E3:14_2_1_1_1_3_3_1"/>
<font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c1">Gc</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:14_2_1_1_1_3_3_1"/><b>then </b><i><font color="Green" title="E25">A19</font></i>: 
 ex <font color="Olive" title="b1">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">i2</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T25" title="GR_CY_1:th.25">GR_CY_1:25</a></i>;<br/>
<a NAME="E5:14_2_1_1_1_3_3_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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" title="c10">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:14_2_1_1_1_3_3_1_1"/><i><font color="Green" title="E26">A20</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">n</font>
 <b>and </b><br/><a NAME="E3:14_2_1_1_1_3_3_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>and </b><br/><a NAME="E4:14_2_1_1_1_3_3_1_1"/><i><font color="Green" title="E27">A21</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:14_2_1_1_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1_3_3_1"><i><font color="Green" title="E25">A19</font></i></a></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" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E28">A22</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:14_2_1_1_1_3_3_1_1"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="group_1.html#T43" title="GROUP_1:th.43">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" title="E29">A23</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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" title="E30">A24</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E31">A25</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1_3"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1_3_3_1_1_1_1_1"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:14_2_1_1_1_3_3_1_1_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1_3"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:14_2_1_1_1_3_3_1_1_1_1_1"><i><font color="Green" title="E28">A22</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1_3_3_1_1_1_1_1"><i><font color="Green" title="E30">A24</font></i></a></i>;<br/><b>hence </b><a NAME="E7:14_2_1_1_1_3_3_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:14_2_1_1_1_3_3_1_1_1_1_1"><i><font color="Green" title="E31">A25</font></i></a></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" title="E28">A26</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="E29">A27</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:14_2_1_1_1_3_3_1_1_1_1_2"><i><font color="Green" title="E28">A26</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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" title="E30">A28</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1_3_3_1_1"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E31">A29</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1_3"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1_3_3_1_1"><i><font color="Green" title="E27">A21</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:14_2_1_1_1_3_3_1_1_1_1_2"><i><font color="Green" title="E29">A27</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:14_2_1_1_1_3_3_1_1_1_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1_3"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:14_2_1_1_1_3_3_1_1"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:14_2_1_1_1_3_3_1_1_1_1_2"><i><font color="Green" title="E30">A28</font></i></a></i>;<br/><b>hence </b><a NAME="E6:14_2_1_1_1_3_3_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_1_3_3_1_1_1_1_2"><i><font color="Green" title="E31">A29</font></i></a></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" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c7">F</font>
;<br/>

<b>thus </b><a NAME="E8:14_2_1_1_1_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E6:14_2_1_1_1_3"><i><font color="Green" title="E23">A16</font></i></a>, <a class="txt" href="gr_cy_2.html#E7:14_2_1_1_1_3"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c1">Gc</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>
 <b>by </b><i><a class="ref" href="group_1.html#D14" title="GROUP_1:def.14">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" title="c4">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><a NAME="E2:14_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 ;<br/><a NAME="E3:14_2_1_1_2"/><b>then </b><i><font color="Green" title="E16">A30</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><b>set </b><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font>;<br/><a NAME="E4:14_2_1_1_2"/><i><font color="Green" title="E17">A31</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">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" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c4">i1</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c4">i1</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:14_2"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="group_1.html#T70" title="GROUP_1:th.70">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" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E3:14_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>;<br/>
<b>hence </b><a NAME="E4:14_2_1_1_2_1"/>
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E5:14_2_1_1_2"/><i><font color="Green" title="E18">A32</font></i>: 
for <font color="Olive" title="b1">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">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" title="c6">i2</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<i><font color="Green" title="E19">A33</font></i>: <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">i2</font> = 
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2"><i><font color="Green" title="E16">A30</font></i></a>, <a class="ref" href="newton.html#T80" title="NEWTON:th.80">NEWTON:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2"><i><font color="Green" title="E17">A31</font></i></a>, <a class="ref" href="group_1.html#T67" title="GROUP_1:th.67">GROUP_1:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T61" title="GROUP_1:th.61">GROUP_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E2:14_2_1_1_2_2"/>
<font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2"><i><font color="Green" title="E16">A30</font></i></a>, <a class="ref" href="newton.html#T78" title="NEWTON:th.78">NEWTON:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c6">i2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">m</font>
;<br/>

<b>thus </b><a NAME="E4:14_2_1_1_2_2"/>
( <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2"><i><font color="Green" title="E16">A30</font></i></a>, <a class="txt" href="gr_cy_2.html#E1:14_2_1_1_2_2"><i><font color="Green" title="E19">A33</font></i></a>, <a class="ref" href="newton.html#T79" title="NEWTON:th.79">NEWTON:79</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2"><i><font color="Green" title="E16">A30</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E7:14_2_1_1_2"/>
 ex <font color="Olive" title="b1">F</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> = <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c7">F</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E2:14_2_1_1_2_3"/><i><font color="Green" title="E19">A34</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 <b>and </b><br/><a NAME="E3:14_2_1_1_2_3"/><i><font color="Green" title="E20">A35</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> holds <br/><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">p</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E4:14_2_1_1_2_3"/><i><font color="Green" title="E21">A36</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:14_2_1_1_2_3"><i><font color="Green" title="E19">A34</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:14_2_1_1_2_3"/><i><font color="Green" title="E22">A37</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">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" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_2_1_1_2_3_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_2_1_1_2_3_1"/><i><font color="Green" title="E23">A38</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>and </b><br/><a NAME="E4:14_2_1_1_2_3_1"/><i><font color="Green" title="E24">A39</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2_3_1"><i><font color="Green" title="E23">A38</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">n</font>
;<br/>

<b>thus </b><a NAME="E6:14_2_1_1_2_3_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2_3"><i><font color="Green" title="E20">A35</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2_3"><i><font color="Green" title="E21">A36</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2_3_1"><i><font color="Green" title="E23">A38</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2_3_1"><i><font color="Green" title="E24">A39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:14_2_1_1_2_3"/><i><font color="Green" title="E23">A40</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_2_1_1_2_3_2_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:14_2_1_1_2_3_2_1"/><i><font color="Green" title="E24">A41</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:14_2_1_1_2_3"><i><font color="Green" title="E22">A37</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:14_2_1_1_2_3_2_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2_3_2_1"><i><font color="Green" title="E24">A41</font></i></a></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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:14_2_1_1_2_3"/><i><font color="Green" title="E24">A42</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">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" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">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" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_2_1_1_2_3_3_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font> ;<br/>
<a NAME="E3:14_2_1_1_2_3_3_1"/>
<font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c1">Gc</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:14_2_1_1_2_3_3_1"/><b>then </b><i><font color="Green" title="E25">A43</font></i>: 
 ex <font color="Olive" title="b1">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">i2</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T25" title="GR_CY_1:th.25">GR_CY_1:25</a></i>;<br/>
<a NAME="E5:14_2_1_1_2_3_3_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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" title="c10">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:14_2_1_1_2_3_3_1_1"/><i><font color="Green" title="E26">A44</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">n</font>
 <b>and </b><br/><a NAME="E3:14_2_1_1_2_3_3_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>and </b><br/><a NAME="E4:14_2_1_1_2_3_3_1_1"/><i><font color="Green" title="E27">A45</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">i1</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:14_2_1_1_2"><i><font color="Green" title="E18">A32</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2_3_3_1"><i><font color="Green" title="E25">A43</font></i></a></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" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E28">A46</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2"><i><font color="Green" title="E17">A31</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:14_2_1_1_2_3_3_1_1"><i><font color="Green" title="E26">A44</font></i></a>, <a class="ref" href="group_1.html#T43" title="GROUP_1:th.43">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" title="E29">A47</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2"><i><font color="Green" title="E16">A30</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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" title="E30">A48</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E31">A49</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2_3"><i><font color="Green" title="E21">A36</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2_3_3_1_1_1_1_1"><i><font color="Green" title="E29">A47</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:14_2_1_1_2_3_3_1_1_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2_3"><i><font color="Green" title="E20">A35</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:14_2_1_1_2_3_3_1_1_1_1_1"><i><font color="Green" title="E28">A46</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2_3_3_1_1_1_1_1"><i><font color="Green" title="E30">A48</font></i></a></i>;<br/><b>hence </b><a NAME="E7:14_2_1_1_2_3_3_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:14_2_1_1_2_3_3_1_1_1_1_1"><i><font color="Green" title="E31">A49</font></i></a></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" title="E28">A50</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="E29">A51</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:14_2_1_1_2_3_3_1_1_1_1_2"><i><font color="Green" title="E28">A50</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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" title="E30">A52</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2_3_3_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E31">A53</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2_3"><i><font color="Green" title="E21">A36</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2_3_3_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:14_2_1_1_2_3_3_1_1_1_1_2"><i><font color="Green" title="E29">A51</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:14_2_1_1_2_3_3_1_1_1_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2_3"><i><font color="Green" title="E20">A35</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:14_2_1_1_2_3_3_1_1"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:14_2_1_1_2_3_3_1_1_1_1_2"><i><font color="Green" title="E30">A52</font></i></a></i>;<br/><b>hence </b><a NAME="E6:14_2_1_1_2_3_3_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:14_2_1_1_2_3_3_1_1_1_1_2"><i><font color="Green" title="E31">A53</font></i></a></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" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c7">F</font>
;<br/>

<b>thus </b><a NAME="E8:14_2_1_1_2_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E6:14_2_1_1_2_3"><i><font color="Green" title="E23">A40</font></i></a>, <a class="txt" href="gr_cy_2.html#E7:14_2_1_1_2_3"><i><font color="Green" title="E24">A42</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c1">Gc</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>
 <b>by </b><i><a class="ref" href="group_1.html#D14" title="GROUP_1:def.14">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" title="c1">Gc</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
( <font color="Maroon" title="c1">Gc</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a> iff  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i1</font> &amp; <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">i1</font> ) )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:14"><i><font color="Green" title="E14">A2</font></i></a></i>;<br/>


</div>
