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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cohsp_1.html#D7" title="COHSP_1:def.7">COHSP_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:31_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:31_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c5">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><span class="kw"> such that </span><br/><a NAME="E2:31_1_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Z1</font>
 <span class="kw">and </span><br/><a NAME="E3:31_1_1"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c5">Z1</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">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><span class="kw"> such that </span><br/><a NAME="E5:31_1_1"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z</font>
 <span class="kw">and </span><br/><a NAME="E6:31_1_1"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c6">Z</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>;<br/>
<a NAME="E7:31_1_1"/>
<font color="Maroon" title="c6">Z</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Z1</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E3:31_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="waybel_0.html#T27" title="WAYBEL_0:th.27">WAYBEL_0:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:31_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E2:31_1_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c5">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><span class="kw"> such that </span><br/><a NAME="E2:31_1_2"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Z1</font>
 <span class="kw">and </span><br/><a NAME="E3:31_1_2"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<font color="Maroon" title="c5">Z1</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">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><span class="kw"> such that </span><br/><a NAME="E5:31_1_2"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z</font>
 <span class="kw">and </span><br/><a NAME="E6:31_1_2"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c6">Z</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>;<br/>
<a NAME="E7:31_1_2"/>
<font color="Maroon" title="c6">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Z1</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_1_2"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E3:31_1_2"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="waybel_0.html#T27" title="WAYBEL_0:th.27">WAYBEL_0:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:31_1_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1_2"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E2:31_1_2"><span class="lab"><font color="Green" title="E26">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:31_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice7.html#D8" target="_self" title="LATTICE7:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
