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

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



<b>assume </b><a NAME="E1:12"/>
<font color="Maroon">b</font> <a href="group_1.html#NR2" target="_self">is_of_order_0</a> 
 ;<br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green">E34</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> holds <br/><font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="group_1.html#D11">GROUP_1:def 11</a></i>;<br/>
<a NAME="E3:12"/>
<font color="Maroon">b</font> <a href="group_1.html#K3">"</a>  <a href="group_1.html#NR2" target="_self">is_of_order_0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> holds <br/><font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:12_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:12_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T51">GROUP_1:51</a></i>;<br/>
<a NAME="E3:12_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K3">"</a> 
 
<b>by </b><i><a class="ref" href="group_1.html#T16">GROUP_1:16</a></i>;<br/>
<a NAME="E4:12_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T17">GROUP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E5:12_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1"/>
<font color="Maroon">b</font> <a href="group_1.html#K3">"</a>  <a href="group_1.html#NR2" target="_self">is_of_order_0</a> 
 <b>by </b><i><a class="ref" href="group_1.html#D11">GROUP_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:12"/>
<font color="Maroon">b</font> <a href="group_1.html#K3">"</a>  <a href="group_1.html#NR2" target="_self">is_of_order_0</a> 
 ;<br/>


</div>
