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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V3">Group-like</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="group_7.html#K24" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="group_7.html#K24" target="_self">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span>;<br/>
<a NAME="E1:81"/>
<span class="p1"><a href="group_7.html#K24" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="group_7.html#K24" target="_self">*&gt;</a></span> 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="81_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="group_7.html#K24" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="group_7.html#K24" target="_self">*&gt;</a></span> <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>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="group_7.html#K24" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="group_7.html#K24" target="_self">*&gt;</a></span> as    <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> ;<br/>
<a NAME="E3:81"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V3">Group-like</a>  <a href="group_1.html#L1">HGrStr</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:81_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></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="81_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i><a class="txt" href="group_7.html#E1:81_2"><i><font color="Green">E35</font></i></a>, <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="81_2_1_1"><b>suppose </b></a><a NAME="E1:81_2_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V3">Group-like</a>  <a href="group_1.html#L1">HGrStr</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/><a NAME="E3:81_2_1_1"/>
<span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:81_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><b>hence </b><a NAME="E4:81_2_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="group_7.html#E1:81_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_2_1_2"><b>suppose </b></a><a NAME="E1:81_2_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V3">Group-like</a>  <a href="group_1.html#L1">HGrStr</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/><a NAME="E3:81_2_1_2"/>
<span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:81_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><b>hence </b><a NAME="E4:81_2_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="group_7.html#E1:81_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_2_1_3"><b>suppose </b></a><a NAME="E1:81_2_1_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V3">Group-like</a>  <a href="group_1.html#L1">HGrStr</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/><a NAME="E3:81_2_1_3"/>
<span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:81_2_1_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><b>hence </b><a NAME="E4:81_2_1_3"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="group_7.html#E1:81_2_1_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:81"/>
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <span class="p2"><a href="group_7.html#K19" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_7.html#K19" target="_self">*&gt;</a></span></span>)</span> <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"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></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#T8" target="_self">Th8</a></i>;<br/>


</div>
