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

<b>let </b><font color="Maroon" title="c1">Gc</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">k</font> implies  ex <font color="Olive" title="b1">g1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font> st <br/>(  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; ( for <font color="Olive" title="b2">g2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>  st  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/><font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> ) ) )</span><br/>

<b>set </b><font color="Maroon" title="c2">n</font> =  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>;<br/>
<b>given </b><font color="Maroon" title="c3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:22"/><i><font color="Green" title="E18">A1</font></i>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font> st <br/>(  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; ( for <font color="Olive" title="b2">g2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>  st  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/><font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> ) )</span><br/>

<b>consider </b><font color="Maroon" title="c4">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font><b> such that </b><br/><a NAME="E4:22"/><i><font color="Green" title="E19">A2</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="c4">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#D9" title="GR_CY_1:def.9">GR_CY_1:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; ( for <font color="Olive" title="b1">g2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>  st  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g2</font> ) )</span><br/>

<i><font color="Green" title="E20">A3</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> 2 = 
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T29" title="GR_CY_1:th.29">GR_CY_1:29</a></i>
;<br/>
<a NAME="E6:22"/><i><font color="Green" title="E21">A4</font></i>: 
not <font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">being_of_order_0</a>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T26" title="GR_CY_1:th.26">GR_CY_1:26</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green" title="E22">A5</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 
<b>by </b><i><a class="txt" href="#E4:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></i>;<br/>
<a NAME="E8:22"/><i><font color="Green" title="E23">A6</font></i>: 
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 
<b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<a NAME="E9:22"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
;<br/>
<a NAME="E10:22"/>
for <font color="Olive" title="b1">p</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  holds <br/>2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_2"/><i><font color="Green" title="E25">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font>
 <b>and </b><br/><a NAME="E2:22_2"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>and </b><br/><a NAME="E3:22_2"/><i><font color="Green" title="E27">A10</font></i>: 
2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:22_2"/><i><font color="Green" title="E28">A11</font></i>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E9:22"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="#E3:22_2"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E5:22_2"/><i><font color="Green" title="E29">A12</font></i>: 
not <font color="Maroon" title="c4">g</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">being_of_order_0</a>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T26" title="GR_CY_1:th.26">GR_CY_1:26</a></i>;<br/>
<a NAME="E6:22_2"/><i><font color="Green" title="E30">A13</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="#E8:22"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="#E2:22_2"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E7:22_2"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:22_2"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>;<br/>
<b>hence </b><a NAME="E8:22_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:22"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="#E4:22_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="#E5:22_2"><i><font color="Green" title="E29">A12</font></i></a>, <a class="txt" href="#E6:22_2"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:22"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="#E5:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="#E6:22"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>  st  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g2</font></span><br/>

<b>let </b><font color="Maroon" title="c5">g2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E12:22"/><i><font color="Green" title="E25">A14</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>and </b><br/><a NAME="E13:22"/><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c6">k1</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="E15:22"/><i><font color="Green" title="E27">A16</font></i>: 
<font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">k1</font>
 <b>by </b><i><a class="txt" href="#E4:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="gr_cy_2.html#T12" target="_self" title="GR_CY_2:th.12">Th12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c7">t</font>, <font color="Maroon" title="c8">t1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:22_3"/><i><font color="Green" title="E28">A17</font></i>: 
<font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">t1</font>
 <b>and </b><br/><a NAME="E3:22_3"/><i><font color="Green" title="E29">A18</font></i>: 
<font color="Maroon" title="c8">t1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="#E9:22"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="nat_1.html#T17" title="NAT_1:th.17">NAT_1:17</a></i>;<br/><a NAME="E4:22_3"/>
<font color="Maroon" title="c8">t1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_3_1"/>
<font color="Maroon" title="c8">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then </b><i><font color="Green" title="E30">A19</font></i>: <font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">k1</font> = 
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:22_3"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48" title="GROUP_1:th.48">GROUP_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42" title="GROUP_1:th.42">GROUP_1:42</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_3_1_2"/>
( <font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1_2_1"><b>suppose </b></a><a NAME="E1:22_3_1_2_1"/>
<font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:22_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font>
 <b>by </b><i><a class="txt" href="#E2:22_3_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="group_1.html#T43" title="GROUP_1:th.43">GROUP_1:43</a></i>;<br/><b>hence </b><a NAME="E3:22_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E12:22"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="#E15:22"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="group_1.html#T84" title="GROUP_1:th.84">GROUP_1:84</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1_2_2"><b>suppose </b></a><a NAME="E1:22_3_1_2_2"/>
<font color="Maroon" title="c7">t</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:22_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E13:22"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="#E15:22"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="#E2:22_3_1"><i><font color="Green" title="E30">A19</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:22_3"/><b>then </b><i><font color="Green" title="E30">A20</font></i>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">t1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><a NAME="E6:22_3"/><i><font color="Green" title="E31">A21</font></i>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">t1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 <b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E3:22_3"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E7:22_3"/><i><font color="Green" title="E32">A22</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">t1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_2"><b>proof </b></a><div class="add">

<b>thus </b> <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> 2
<b>by </b><i><a class="txt" href="#E12:22"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="#E15:22"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="group_1.html#T82" title="GROUP_1:th.82">GROUP_1:82</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">t1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="#E2:22_3"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E18">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">t</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48" title="GROUP_1:th.48">GROUP_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:22"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T82" title="GROUP_1:th.82">GROUP_1:82</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42" title="GROUP_1:th.42">GROUP_1:42</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">t1</font></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>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E8:22_3"/>
not <font color="Maroon" title="c4">g</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">being_of_order_0</a>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T26" title="GR_CY_1:th.26">GR_CY_1:26</a></i>;<br/><b>hence </b><a NAME="E9:22_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:22"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="#E5:22_3"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="#E6:22_3"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="#E7:22_3"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:22"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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