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

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

<b>set </b><font color="Maroon">L</font> =  <a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font>;<br/>
<b>let </b><font color="Maroon">Y</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E1:15"/><i><font color="Green">E21</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">In</font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> as   <a href="relset_1.html#NM3">Relation</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E4:15"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">In</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">y</font> = <font color="Maroon">x'</font>;<br/>

<a NAME="E3:15_1"/>
<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
;<br/>
<b>hence </b><a NAME="E4:15_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">In</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:15"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">In</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T22">RELSET_1:22</a></i>;<br/>
<a NAME="E6:15"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">In</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">x</font> = <font color="Maroon">y'</font>;<br/>

<a NAME="E3:15_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font>
 
;<br/>
<b>hence </b><a NAME="E4:15_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">In</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:15"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">In</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T23">RELSET_1:23</a></i>;<br/>
<a NAME="E8:15"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:15_3"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font>, <font color="Maroon">Z'</font> = <font color="Maroon">Z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:15_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> implies <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<a NAME="E2:15_3_1"/><b>then </b>
<font color="Maroon">Y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Z'</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:15_3_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:15_3"/>
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_3_2"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:15_3_2"/><b>then </b>
<font color="Maroon">Y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Z'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:15_3_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:15"/><b>then </b><i><font color="Green">E28</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K1" target="_self">RelIncl</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E11:15"/>
 <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">Y'</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">Y'</font></span>)</span> #) is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T57">YELLOW_0:57</a></i>;<br/>
<a NAME="E12:15"/><b>then </b>
 <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">Y'</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" target="_self">RelIncl</a> <font color="Maroon">Y'</font></span>)</span> #) is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="wellord2.html#T8">WELLORD2:8</a></i>;<br/>
<b>hence </b><a NAME="E13:15"/>
 <a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">Y</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font>
 ;<br/>


</div>
