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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V6">finite</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E1:31"/>
 <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="cohsp_1.html#R1">includes_lattice_of</a>  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="cohsp_1.html#D8">COHSP_1:def 8</a><br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:31_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> : <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice7.html#E1:31_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V12">lower</a> )
 ;<br/>
<a NAME="E4:31_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> : <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice7.html#E1:31_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:31_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V12">lower</a> )
 ;<br/>
<a NAME="E7:31_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V12">lower</a>
 
<b>by </b><i><a class="txt" href="lattice7.html#E3:31_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="lattice7.html#E6:31_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel_0.html#T27">WAYBEL_0:27</a></i>;<br/>
<a NAME="E8:31_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> : <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E9:31_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice7.html#E3:31_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="lattice7.html#E6:31_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:31_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> : <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice7.html#E1:31_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:31_1_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V12">lower</a> )
 ;<br/>
<a NAME="E4:31_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> : <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice7.html#E1:31_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:31_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V12">lower</a> )
 ;<br/>
<a NAME="E7:31_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V12">lower</a>
 
<b>by </b><i><a class="txt" href="lattice7.html#E3:31_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="lattice7.html#E6:31_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel_0.html#T27">WAYBEL_0:27</a></i>;<br/>
<a NAME="E8:31_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> : <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E9:31_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice7.html#E3:31_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="lattice7.html#E6:31_1_2"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="lattice7.html#E2:31_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:31"/>
 <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is    <a href="lattice7.html#M2" target="_self">Ring_of_sets</a> 
 <b>by </b><i><a class="ref" href="lattice7.html#D8" target="_self">Def8</a></i>;<br/>


</div>
