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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V2">upper-bounded</a> <a href="lattice3.html#V2">with_infima</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E1:8"/><i><font color="Green">E9</font></i>: 
 <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>;<br/>
<a NAME="E2:8"/><i><font color="Green">E10</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<a NAME="E3:8"/><i><font color="Green">E11</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<a NAME="E4:8"/><b>then </b><i><font color="Green">E12</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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E5:8"/><i><font color="Green">E13</font></i>: 
 <a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>
<b>thus </b><a NAME="E4:8_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:8_1"><i><font color="Green">E14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:8"/><i><font color="Green">E14</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:8_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:8_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:8_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel22.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel22.html#E3:8_2"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:8"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:8_3"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:8_3"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:8_3"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel22.html#E1:8_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="waybel22.html#E3:8_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:8_3"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E4:8"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:8_3"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E4:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="waybel22.html#E4:8_3"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E9:8_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E4:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="waybel22.html#E1:8_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="waybel22.html#E3:8_3"><i><font color="Green">E16</font></i></a>, <a class="txt" href="waybel22.html#E4:8_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel22.html#E6:8_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel22.html#E8:8_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:8_3"/>
<font color="Maroon">c<sub>9</sub></font> <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">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel22.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel22.html#E3:8_3"><i><font color="Green">E16</font></i></a>, <a class="txt" href="waybel22.html#E6:8_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel22.html#E8:8_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as    <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel22.html#E6:8"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:8_4"/><i><font color="Green">E15</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="wellord1.html#D6">WELLORD1:def 6</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:8_4_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:8_4_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:8_4_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8_4_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="waybel22.html#E3:8_4_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:8_4_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E4:8"><i><font color="Green">E12</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:8_4_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E4:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="waybel22.html#E4:8_4_1"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E9:8_4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E4:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="waybel22.html#E1:8_4_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="waybel22.html#E3:8_4_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel22.html#E4:8_4_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel22.html#E6:8_4_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="waybel22.html#E8:8_4_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><a NAME="E10:8_4_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <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">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel22.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel22.html#E3:8_4_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel22.html#E6:8_4_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="waybel22.html#E8:8_4_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E11:8_4_1"/>
<font color="Maroon">c<sub>10</sub></font> <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">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8_4"><i><font color="Green">E15</font></i></a>, <a class="txt" href="waybel22.html#E1:8_4_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E3:8_4"/>
<font color="Maroon">c<sub>10</sub></font> <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">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E4:8_4"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <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">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8_4"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:8_4"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:8_4"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E6:8_4"><i><font color="Green">E17</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:8_4"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E6:8_4"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E11:8_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel22.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel22.html#E4:8_4"><i><font color="Green">E16</font></i></a>, <a class="txt" href="waybel22.html#E6:8_4"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel22.html#E8:8_4"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel22.html#E10:8_4"><i><font color="Green">E19</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E12:8_4"/>
<font color="Maroon">c<sub>10</sub></font> <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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E6:8_4"><i><font color="Green">E17</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:8"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>;<br/>
<a NAME="E12:8"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>10</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D18">YELLOW_0:def 18</a><br/>

<b>assume </b><a NAME="E1:8_5"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>11</sub></font>, <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_5_1"/>
( <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_5_1_1"><b>suppose </b></a><a NAME="E1:8_5_1_1"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E16</font></i>:  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="yellow_1.html#T19">YELLOW_1:19</a></i>
;<br/><a NAME="E3:8_5_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/><b>hence </b><a NAME="E4:8_5_1_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E2:8_5_1_1"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_5_1_2"><b>suppose </b></a><a NAME="E1:8_5_1_2"/><i><font color="Green">E16</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel22.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel22.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E5:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E5:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:8_5_1_2"/><i><font color="Green">E17</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8_5_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow_1.html#T20">YELLOW_1:20</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel22.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>;<br/><a NAME="E6:8_5_1_2"/>
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> <font color="Maroon">c<sub>11</sub></font> holds <br/> <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_5_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_5_1_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:8_5_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:8_5_1_2_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>
<b>thus </b><a NAME="E5:8_5_1_2_1"/>
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E4:8_5_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="waybel_4.html#T22">WAYBEL_4:22</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:8_5_1_2"/><b>then </b><i><font color="Green">E18</font></i>: 
not <font color="Maroon">c<sub>15</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8_5_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="waybel22.html#E4:8_5_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E8:8_5_1_2"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V2">filtered</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_5_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:8_5_1_2_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E2:8_5_1_2_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E1:8_5_1_2_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:8_5_1_2_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>
<b>thus </b><a NAME="E5:8_5_1_2_2"/>
( <font color="Maroon">c<sub>16</sub></font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="waybel_0.html#V2">filtered</a> )
 <b>by </b><i><a class="txt" href="waybel22.html#E4:8_5_1_2_2"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:8_5_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V13">upper</a>
 ;<br/><a NAME="E10:8_5_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i><a class="txt" href="waybel22.html#E4:8_5_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="yellow_2.html#T39">YELLOW_2:39</a></i>;<br/><a NAME="E11:8_5_1_2"/>
<font color="Maroon">c<sub>15</sub></font> is <a href="waybel_0.html#V2">filtered</a>
 <b>by </b><i><a class="txt" href="waybel22.html#E4:8_5_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel22.html#E8:8_5_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="yellow_2.html#T41">YELLOW_2:41</a></i>;<br/><b>hence </b><a NAME="E12:8_5_1_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E7:8_5_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel22.html#E10:8_5_1_2"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:8"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>10</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D4">WAYBEL_0:def 4</a><br/>

<b>assume </b><a NAME="E1:8_6"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>11</sub></font>, <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel22.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel22.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E5:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E5:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:8_6"/><i><font color="Green">E17</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T21">YELLOW_1:21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel22.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:8_6"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8_6"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E8:8_6"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E7:8_6"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:8_6"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>
<a NAME="E11:8_6"/>
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E10:8_6"><i><font color="Green">E19</font></i></a>, <a class="ref" href="waybel_4.html#T22">WAYBEL_4:22</a></i>;<br/>
<a NAME="E12:8_6"/><b>then </b><i><font color="Green">E20</font></i>: 
not <font color="Maroon">c<sub>15</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="waybel22.html#E4:8_6"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel22.html#E7:8_6"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E13:8_6"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V2">filtered</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:8_6_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E2:8_6_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:8_6_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>
<b>thus </b><a NAME="E5:8_6_1"/>
( <font color="Maroon">c<sub>18</sub></font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Maroon">c<sub>18</sub></font> is <a href="waybel_0.html#V2">filtered</a> )
 <b>by </b><i><a class="txt" href="waybel22.html#E4:8_6_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:8_6"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V13">upper</a>
 
;<br/>
<a NAME="E15:8_6"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is <a href="waybel_0.html#V13">upper</a>
 
<b>by </b><i><a class="txt" href="waybel22.html#E4:8_6"><i><font color="Green">E17</font></i></a>, <a class="ref" href="waybel_0.html#T28">WAYBEL_0:28</a></i>;<br/>
<a NAME="E16:8_6"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V2">filtered</a>
 
<b>by </b><i><a class="txt" href="waybel22.html#E13:8_6"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E17:8_6"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:8_6_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font><b> such that </b><br/><a NAME="E4:8_6_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="waybel22.html#E1:8_6_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<a NAME="E5:8_6_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E4:8_6_2"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:8_6_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/>

<b>thus </b><a NAME="E8:8_6_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E4:8_6_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel22.html#E7:8_6_2"><i><font color="Green">E26</font></i></a></i>;<br/>

<a NAME="E9:8_6_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel22.html#E4:8_6_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E10:8_6_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E7:8_6_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:8_6"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> is <a href="waybel_0.html#V2">filtered</a>
 
<b>by </b><i><a class="txt" href="waybel22.html#E4:8_6"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel22.html#E16:8_6"><i><font color="Green">E23</font></i></a>, <a class="ref" href="waybel_0.html#T47">WAYBEL_0:47</a></i>;<br/>
<b>hence </b><a NAME="E19:8_6"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel22.html#E12:8_6"><i><font color="Green">E20</font></i></a>, <a class="txt" href="waybel22.html#E15:8_6"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:8"/>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is   <a href="waybel_5.html#NM3">CLSubFrame</a> of  <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel22.html#E12:8"><i><font color="Green">E15</font></i></a></i>;<br/>


</div>
