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

<b>let </b><font color="Maroon">G1</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>;<br/>



<b>reconsider </b><font color="Maroon">G</font> = <span class="p1"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span> as  <a href="group_7.html#V2" target="_self">Group-like</a> <a href="group_7.html#V3" target="_self">associative</a> <a href="group_7.html#NM1">HGrStr-Family</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">lF</font> = <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span>, <font color="Maroon">p</font> = <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="group_7.html#K10" target="_self">*&gt;</a></span> as    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">G</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E3:82"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">lF</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E4:82"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> is    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:82_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a><br/>


</div><b>end;</b></div>
<a NAME="E5:82"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> holds <br/> ex <font color="Olive">H</font> being  <a href="group_1.html#NM1">Group</a> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">H</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> <a href="group_1.html#K3">"</a>  &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lF</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:82_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:82_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 as   <a href="group_1.html#NM1">Group</a> <b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>take </b>
<font color="Maroon">z</font> <a href="group_1.html#K3">"</a> 
;<br/>

<b>thus </b><a NAME="E5:82_2"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<b>thus </b><a NAME="E6:82_2"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a> 
 <b>by </b><i>, <a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>

<b>thus </b><a NAME="E7:82_2"/>
<font color="Maroon">z</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">lF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:82"/>
<span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="group_7.html#K10" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="group_7.html#D1" target="_self">Def1</a>, </i>;<br/>


</div>
