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

<b>let </b><font color="Maroon" title="c1">G1</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">G</font> = <span class="p1"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span> as  <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a> <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a> <a href="group_7.html#NM1" title="GROUP_7:NM.1">HGrStr-Family</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">lF</font> = <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>, <font color="Maroon" title="c5">p</font> = <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c3">G</font></span>)</span> <b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/>
<a NAME="E3:82"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  &amp; <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c4">lF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E4:82"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c5">p</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a><br/>


</div><b>end;</b></div>
<a NAME="E5:82"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> ex <font color="Olive" title="b2">H</font> being  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ex <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">H</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">lF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:82_2"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:82_2"/><b>then </b><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> <b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">H</font> <b>by </b><i><a class="txt" href="group_7.html#E3:82"><i><font color="Green" title="E35">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">H</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">z</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
;<br/>

<b>thus </b><a NAME="E5:82_2"/>
<font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:82_2"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>

<b>thus </b><a NAME="E6:82_2"/>
<font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 <b>by </b><i><a class="txt" href="group_7.html#E2:82_2"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>;<br/>

<b>thus </b><a NAME="E7:82_2"/>
<font color="Maroon" title="c8">z</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">lF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E3:82"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="group_7.html#E2:82_2"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">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" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="group_7.html#E4:82"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="group_7.html#T10" target="_self" title="GROUP_7:th.10">Th10</a></i>;<br/>


</div>
