<?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">G2</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">G3</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>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G3</font>;<br/>











<b>set </b><font color="Maroon">G</font> = <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon">lF</font> = <span class="p1"><a href="group_7.html#K24" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="group_7.html#K24" target="_self">*&gt;</a></span>, <font color="Maroon">p</font> = <span class="p1"><a href="group_7.html#K24" 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 class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="group_7.html#K24" 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> <span class="p2"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E2:84"/><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">p</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="group_1.html#K3">"</a>  &amp; <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> &amp; <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">G3</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> &amp; <font color="Maroon">lF</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">lF</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E3:84"/><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="enumset1.html#K1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:84_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a>, <a class="ref" href="finseq_3.html#T30">FINSEQ_3:30</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="E4:84"/>
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="enumset1.html#K1">{</a><span class="default">1,2,3</span><a href="enumset1.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> <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span> <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="84_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="E1:84_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="enumset1.html#K1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1"><b>suppose </b></a><a NAME="E1:84_2_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">H</font> = <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span> <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#T62">FINSEQ_1:62</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="E4:84_2_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:84_2_1_1"/>
<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="E6:84_2_1_1"/>
<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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_2"><b>suppose </b></a><a NAME="E1:84_2_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">H</font> = <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 as   <a href="group_1.html#NM1">Group</a> <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 2 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="E4:84_2_1_2"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:84_2_1_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="E6:84_2_1_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_3"><b>suppose </b></a><a NAME="E1:84_2_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">H</font> = <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 3 as   <a href="group_1.html#NM1">Group</a> <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 3 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="E4:84_2_1_3"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K22" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">G3</font></span><a href="group_7.html#K22" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:84_2_1_3"/>
<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="E6:84_2_1_3"/>
<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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:84"/>
<span class="p1"><a href="group_7.html#K24" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="group_7.html#K24" 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#K24" 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 class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="group_7.html#K24" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="group_7.html#D1" target="_self">Def1</a>, </i>;<br/>


</div>
