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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> is    <a href="lattice7.html#M2" title="LATTICE7:mode.2">Ring_of_sets</a> </span><br/>

<b>set </b><font color="Maroon" title="c2">X</font> =  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:31"/>
 <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="cohsp_1.html#R1" title="COHSP_1:pred.1">includes_lattice_of</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</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" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="cohsp_1.html#D8" title="COHSP_1:def.8">COHSP_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> or  not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> or ( <font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E2:31_1"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</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">

<b>consider </b><font color="Maroon" title="c5">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E2:31_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Z</font> &amp; <font color="Maroon" title="c5">Z</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> )
 <b>by </b><i><a class="txt" href="#E1:31_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">Z1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:31_1_1"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z1</font> &amp; <font color="Maroon" title="c6">Z1</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> )
 <b>by </b><i><a class="txt" href="#E1:31_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<a NAME="E5:31_1_1"/>
<font color="Maroon" title="c5">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Z1</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>
 
<b>by </b><i><a class="txt" href="#E2:31_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="#E4:31_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="waybel_0.html#T27" title="WAYBEL_0:th.27">WAYBEL_0:27</a></i>;<br/>
<b>hence </b><a NAME="E6:31_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:31_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="#E4:31_1_1"><i><font color="Green" title="E26">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:31_1"/>
<font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</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">

<b>consider </b><font color="Maroon" title="c5">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E2:31_1_2"/><i><font color="Green" title="E25">A5</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Z</font> &amp; <font color="Maroon" title="c5">Z</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> )
 <b>by </b><i><a class="txt" href="#E1:31_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">Z1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:31_1_2"/><i><font color="Green" title="E26">A6</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z1</font> &amp; <font color="Maroon" title="c6">Z1</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> )
 <b>by </b><i><a class="txt" href="#E1:31_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<a NAME="E5:31_1_2"/>
<font color="Maroon" title="c5">Z</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z1</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>
 
<b>by </b><i><a class="txt" href="#E2:31_1_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="#E4:31_1_2"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="waybel_0.html#T27" title="WAYBEL_0:th.27">WAYBEL_0:27</a></i>;<br/>
<b>hence </b><a NAME="E6:31_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:31_1_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="#E4:31_1_2"><i><font color="Green" title="E26">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1"/>
( <font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:31_1"><i><font color="Green" title="E24">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:31"/>
 <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> is    <a href="lattice7.html#M2" title="LATTICE7:mode.2">Ring_of_sets</a> 
 <b>by </b><i><a class="ref" href="lattice7.html#D8" target="_self" title="LATTICE7:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
