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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">L1</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green">E40</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font>
 ;<br/>

<b>let </b><font color="Maroon">B1</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">B2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font>;<br/>



<b>assume </b><a NAME="E2:9"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:9_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B2</font>
 <b>and </b><br/><a NAME="E2:9_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B2</font>
 ;<br/><a NAME="E3:9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">Y</font>,<font color="Maroon">L1</font></span>)</span> where <font color="Olive">Y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B2</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">L1</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/><b>then consider </b><font color="Maroon">Y1</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B2</font><b> such that </b><br/><a NAME="E5:9_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Y1</font>,<font color="Maroon">L1</font>
 <b>and </b><br/><a NAME="E6:9_1"/><i><font color="Green">E49</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Y1</font>,<font color="Maroon">L1</font>
 ;<br/><a NAME="E7:9_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">Y</font>,<font color="Maroon">L1</font></span>)</span> where <font color="Olive">Y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B2</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">L1</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/><b>then consider </b><font color="Maroon">Y2</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B2</font><b> such that </b><br/><a NAME="E9:9_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Y2</font>,<font color="Maroon">L1</font>
 <b>and </b><br/><a NAME="E10:9_1"/><i><font color="Green">E52</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Y2</font>,<font color="Maroon">L1</font>
 ;<br/><a NAME="E11:9_1"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Y1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font>,<font color="Maroon">L1</font> &amp;  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font></span>)</span>,<font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Y1</font>,<font color="Maroon">L1</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Y2</font>,<font color="Maroon">L1</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="yellow_2.html#T3">YELLOW_2:3</a></i>;<br/><a NAME="E12:9_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">Y</font>,<font color="Maroon">L1</font></span>)</span> where <font color="Olive">Y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B2</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">L1</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E13:9_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/><b>hence </b><a NAME="E14:9_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:9"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B2</font> is <a href="waybel23.html#V2">join-closed</a>
 
<b>by </b><i><a class="ref" href="waybel23.html#T18">WAYBEL23:18</a></i>;<br/>
<a NAME="E5:9"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">L1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a>, <a class="ref" href="yellow_0.html#T42">YELLOW_0:42</a></i>;<br/>
<a NAME="E6:9"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">L1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">Y</font>,<font color="Maroon">L1</font></span>)</span> where <font color="Olive">Y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B2</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">L1</font> </span>}</span> 
 
;<br/>
<a NAME="E7:9"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">L1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/>
<a NAME="E8:9"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B2</font>
 
;<br/>
<a NAME="E9:9"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>  st not <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> holds <br/> ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B2</font> &amp; not <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/>

<a NAME="E1:9_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> )
 
;<br/>
<a NAME="E2:9_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>assume </b><a NAME="E4:9_2"/>
not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E5:9_2"/><b>then </b>
not <font color="Maroon">y</font> <a href="tarski.html#R1">c=</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>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:9_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E8:9_2"/><i><font color="Green">E57</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:9_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E11:9_2"/>
<font color="Maroon">y1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E13:9_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E14:9_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E15:9_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y1</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<a NAME="E17:9_2"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:9_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E19:9_2"/><i><font color="Green">E87</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">b</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T38">YELLOW_0:38</a></i>;<br/>
<a NAME="E20:9_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<a NAME="E21:9_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">Y</font>,<font color="Maroon">L1</font></span>)</span> where <font color="Olive">Y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B2</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">L1</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E22:9_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/>

<a NAME="E23:9_2"/>
not <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E24:9_2"/>
not <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</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>thus </b><a NAME="E25:9_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:9"/>
 <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B2</font> is  <a href="waybel23.html#V6">with_bottom</a>  <a href="waybel23.html#M1">CLbasis</a> of <font color="Maroon">L1</font>
 <b>by </b><i>, , <a class="ref" href="waybel23.html#T49">WAYBEL23:49</a>, <a class="ref" href="waybel23.html#D8">WAYBEL23:def 8</a></i>;<br/>


</div>
