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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</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"> for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">b</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">being_of_order_0</a> holds <br/><font color="Olive" title="b1">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  is <a href="group_1.html#V4" title="GROUP_1:attr.4">being_of_order_0</a></span><br/><b>let </b><font color="Maroon" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">b</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">being_of_order_0</a> implies <font color="Maroon" title="c2">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  is <a href="group_1.html#V4" title="GROUP_1:attr.4">being_of_order_0</a> )</span><br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c2">b</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">being_of_order_0</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  is <a href="group_1.html#V4" title="GROUP_1:attr.4">being_of_order_0</a></span><br/>

<a NAME="E2:9"/>
for <font color="Olive" title="b1">n</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="c2">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">n</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">G</font> holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</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="c2">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">n</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">G</font> implies <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:9_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">n</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">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:9_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <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">G</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T51" title="GROUP_1:th.51">GROUP_1:51</a></i>;<br/>
<a NAME="E3:9_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<b>by </b><i><a class="ref" href="group_1.html#T16" title="GROUP_1:th.16">GROUP_1:16</a></i>;<br/>
<a NAME="E4:9_1"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">n</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">G</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T17" title="GROUP_1:th.17">GROUP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="group_1.html#D11" title="GROUP_1:def.11">GROUP_1:def 11</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="E3:9"/>
<font color="Maroon" title="c2">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  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="group_1.html#D11" title="GROUP_1:def.11">GROUP_1:def 11</a></i>; <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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