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

<b>let </b><font color="Maroon">R</font> be   <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span>;<br/>





<b>set </b><font color="Maroon">I</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">xD</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ;<br/>
<a NAME="E1:7"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:7_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 ;<br/>

<a NAME="E2:7_1"/><b>then </b>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">d</font> &amp; <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
;<br/>
<b>hence </b><a NAME="E3:7_1"/>
<font color="Maroon">a</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">xD</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:7"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T1">WAYBEL_2:1</a></i>;<br/>
<a NAME="E4:7"/><b>then </b>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">xD</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>
<a NAME="E5:7"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">xD</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">xD</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font></span>)</span>
 ;<br/><a NAME="E2:7_2"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><a NAME="E3:7_2"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_4.html#T53">YELLOW_4:53</a></i>;<br/><a NAME="E4:7_2"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E5:7_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:7_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_2.html#T45">YELLOW_2:45</a></i>;<br/><a NAME="E7:7_2"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:7_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:7_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E11:7_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM1">Element</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> ;<br/><a NAME="E13:7_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E14:7_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E15:7_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</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="E7:7"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:7"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E10:7"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E12:7"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E13:7"/>
verum
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM1">Element</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#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span> ;<br/>
<a NAME="E15:7"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T45">YELLOW_2:45</a></i>;<br/>
<a NAME="E16:7"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">xD</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>
<a NAME="E17:7"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font>
 
;<br/>
<a NAME="E18:7"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">xD</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">d</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E19:7"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">d</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">xD</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E20:7"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">xD</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E21:7"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>


</div>
