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

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">X</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">L</font></span>)</span></span>)</span>;<br/>



<a NAME="E1:10"/>
<font color="Maroon">X</font> <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">L</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E2:10"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </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="E1:10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:10_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E3:10_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/><b>then consider </b><font color="Maroon">x1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:10_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 ;<br/><b>thus </b><a NAME="E6:10_1"/>
<font color="Maroon">x</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">L</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:10"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</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">L</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:10_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:10_2"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E3:10_2"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/><b>then consider </b><font color="Maroon">Z1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:10_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z1</font>
 ;<br/><b>thus </b><a NAME="E6:10_2"/>
<font color="Maroon">Z</font> is <a href="waybel_0.html#V12">lower</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">unX</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> as  <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="waybel_0.html#T26">WAYBEL_0:26</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:10"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E9:10"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:10"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/>
<b>then consider </b><font color="Maroon">z1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E12:10"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font>
 ;<br/>
<b>consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:10"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">unX</font> = <font color="Maroon">unX</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:10_3"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:10_3"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E3:10_3"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/><b>then consider </b><font color="Maroon">Z1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:10_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z1</font>
 ;<br/><b>thus </b><a NAME="E6:10_3"/>
<font color="Maroon">Z</font> is <a href="waybel_0.html#V1">directed</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>, <font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:10_4"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:10_4"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font>, <font color="Maroon">Y1</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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">Z1</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">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:10_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E6:10_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">V1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Z1</font>
 <b>and </b><br/><a NAME="E7:10_4"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">Y1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Z1</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/><a NAME="E8:10_4"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><a NAME="E9:10_4"/><b>then </b>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y'</font> where <font color="Olive">Y'</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/><b>then consider </b><font color="Maroon">Z2</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:10_4"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z2</font>
 ;<br/><b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font>;<br/><b>thus </b><a NAME="E13:10_4"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E14:10_4"/>
( <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, , <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>hence </b><a NAME="E15:10_4"/>
<font color="Maroon">V</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">unX</font> = <font color="Maroon">unX</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i>, , <a class="ref" href="waybel_0.html#T46">WAYBEL_0:46</a></i>;<br/>
<b>reconsider </b><font color="Maroon">unX</font> = <font color="Maroon">unX</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">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</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">L</font></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:10_5"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:10_5"/><b>then </b>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E3:10_5"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">unX</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:10"/><b>then </b>
<font color="Maroon">unX</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E22:10"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">unX</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6"><b>now </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="E1:10_6"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> ;<br/><a NAME="E3:10_6"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><a NAME="E4:10_6"/><b>then </b>
<font color="Maroon">Y'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><b>hence </b><a NAME="E5:10_6"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E24:10"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<a NAME="E25:10"/><b>then </b>
<font color="Maroon">unX</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E26:10"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
