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

<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#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">T</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L1</font>;<br/>



<b>consider </b><font color="Maroon">B1</font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:32"/><i><font color="Green">E40</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="waybel23.html#K2">weight</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>
<a NAME="E3:32"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:32_1"/>
<font color="Maroon">z</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#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E4:32_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 ;<br/>
<b>thus </b><a NAME="E5:32_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B0</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">y</font> );<br/>
<a NAME="E5:32"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b><font color="Maroon">y</font> =  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">z</font>;<br/>

<b>thus </b><a NAME="E3:32_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B0</font>
 <b>by </b><i/>;<br/>

<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E4:32_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">z</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:32"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E8:32"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font>
 <b>and </b><br/><a NAME="E9:32"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<a NAME="E10:32"/>
<font color="Maroon">B0</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:32_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B0</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:32_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E4:32_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:32_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E7:32_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E8:32_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:32"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B0</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T28">CARD_1:28</a></i>;<br/>
<b>set </b><font color="Maroon">B2</font> =  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B0</font>;<br/>
<a NAME="E12:32"/><i><font color="Green">E56</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L1</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_4_1"/>
( <font color="Maroon">L1</font> is <a href="group_1.html#V6">finite</a> or  not <font color="Maroon">L1</font> is <a href="group_1.html#V6">finite</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_1_1"><b>suppose </b></a><a NAME="E1:32_4_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">L1</font> is <a href="group_1.html#V6">finite</a>
 ;<br/><div class="add"><a NAME="E2:32_4_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">B0</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B0</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T50">WAYBEL_0:50</a></i>;<br/><a NAME="E3:32_4_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:32_4_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E3:32_4_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, , <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<a NAME="E4:32_4_1_1_1"/><b>then </b>
<font color="Maroon">T</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<a NAME="E5:32_4_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#D5">WAYBEL11:def 5</a></i>;<br/>
<a NAME="E6:32_4_1_1_1"/>
<font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z1</font>
 
;<br/>
<a NAME="E7:32_4_1_1_1"/><b>then </b>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z1</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E9:32_4_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E10:32_4_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E11:32_4_1_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z1</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>
<a NAME="E12:32_4_1_1_1"/>
(  <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z1</font>,<font color="Maroon">T</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">A</font>,<font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E13:32_4_1_1_1"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z1</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>
<a NAME="E14:32_4_1_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T39">WAYBEL_0:39</a></i>;<br/>
<a NAME="E15:32_4_1_1_1"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">A</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z1</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E16:32_4_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E17:32_4_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B0</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:32_4_1_1"/><b>then </b>
 <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E5:32_4_1_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B0</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_1_2"><b>suppose </b></a><a NAME="E1:32_4_1_2"/>
not <font color="Maroon">L1</font> is <a href="group_1.html#V6">finite</a>
 ;<br/><div class="add"><a NAME="E2:32_4_1_2"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E3:32_4_1_2"/><b>then </b>
not <font color="Maroon">T</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E4:32_4_1_2"/><b>then </b>
not <font color="Maroon">B1</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="yellow15.html#T31">YELLOW15:31</a></i>;<br/><a NAME="E5:32_4_1_2"/><b>then </b>
not <font color="Maroon">B0</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:32_4_1_2"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B0</font>
 <b>by </b><i><a class="ref" href="yellow15.html#T28">YELLOW15:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E14:32"/><i><font color="Green">E99</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T42">YELLOW_0:42</a></i>;<br/>
<a NAME="E15:32"/>
 <a href="xboole_0.html#K1">{}</a>  is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B0</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E16:32"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">T</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">T</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">B0</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">T</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E17:32"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B0</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/>
<a NAME="E18:32"/><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">B0</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>
<a NAME="E19:32"/><b>then </b><i><font color="Green">E100</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">B0</font>
 
;<br/>
<b>reconsider </b><font color="Maroon">B2</font> =  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5"><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="E1:32_5"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 <b>and </b><br/><a NAME="E2:32_5"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 ;<br/><a NAME="E3:32_5"/><i><font color="Green">E104</font></i>: 
( <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">T</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">B0</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">T</font> </span>}</span>  &amp; <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">T</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">B0</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">T</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">B0</font><b> such that </b><br/><a NAME="E5:32_5"/><i><font color="Green">E105</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">T</font>
 <b>and </b><br/><a NAME="E6:32_5"/><i><font color="Green">E106</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Y1</font>,<font color="Maroon">T</font>
 ;<br/><b>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">B0</font><b> such that </b><br/><a NAME="E8:32_5"/><i><font color="Green">E109</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">T</font>
 <b>and </b><br/><a NAME="E9:32_5"/><i><font color="Green">E110</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Y2</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/><a NAME="E10:32_5"/><i><font color="Green">E112</font></i>: 
 <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">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/> <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> = 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Y1</font>,<font color="Maroon">T</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">T</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="waybel31.html#E2:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="yellow12.html#T10">YELLOW12:10</a></i>
<br/>.= 
 <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">T</font>
<b>by </b><i>, , <a class="ref" href="yellow_2.html#T3">YELLOW_2:3</a></i>
;<br/><a NAME="E12:32_5"/><b>then </b>
 <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> <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">T</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">B0</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">T</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:32_5"/>
 <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> <font color="Maroon">B2</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B2</font> = <font color="Maroon">B2</font> as  <a href="waybel23.html#V2">join-closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> <b>by </b><i><a class="ref" href="waybel23.html#T18">WAYBEL23:18</a></i>;<br/>
<a NAME="E23:32"/>
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 <font color="Olive">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">y</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> <font color="Maroon">B2</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_6"><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/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:32_6"/>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:32_6"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/>
<a NAME="E4:32_6"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="ref" href="yellow12.html#T13">YELLOW12:13</a></i>;<br/>
<a NAME="E5:32_6"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T36">WAYBEL11:36</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:32_6"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E8:32_6"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E9:32_6"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">u</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x1</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> =  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<a NAME="E11:32_6"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:32_6"/>
<font color="Maroon">B0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T50">WAYBEL_0:50</a></i>;<br/>
<b>hence </b><a NAME="E13:32_6"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 <b>by </b><i/>;<br/>

<a NAME="E14:32_6"/><i><font color="Green">E118</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x1</font>,<font color="Maroon">T</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">u</font>,<font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E15:32_6"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T11">WAYBEL_3:11</a></i>;<br/>
<a NAME="E16:32_6"/><b>then </b>
<font color="Maroon">u</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E17:32_6"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x1</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E2:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>
<a NAME="E18:32_6"/><b>then </b>
<font color="Maroon">x1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T39">WAYBEL_0:39</a></i>;<br/>
<b>hence </b><a NAME="E19:32_6"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>

<a NAME="E20:32_6"/>
<font color="Maroon">u</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="yellow_8.html#T19">YELLOW_8:19</a></i>;<br/>
<a NAME="E21:32_6"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y1</font>
 
<b>by </b><i>, <a class="ref" href="waybel14.html#T26">WAYBEL14:26</a></i>;<br/>
<b>hence </b><a NAME="E22:32_6"/>
<font color="Maroon">b</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="waybel_8.html#T8">WAYBEL_8:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:32"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">B2</font> is    <a href="waybel23.html#M1">CLbasis</a> of <font color="Maroon">L1</font>
 
<b>by </b><i>, <a class="ref" href="waybel23.html#T47">WAYBEL23:47</a></i>;<br/>
<a NAME="E25:32"/>
<font color="Maroon">B2</font> is <a href="waybel23.html#V6">with_bottom</a>
 
<b>by </b><i>, <a class="ref" href="waybel23.html#D8">WAYBEL23:def 8</a></i>;<br/>
<a NAME="E26:32"/><b>then </b>
 <a href="waybel31.html#K1" target="_self">CLweight</a> <font color="Maroon">L1</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B0</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E27:32"/>
 <a href="waybel31.html#K1" target="_self">CLweight</a> <font color="Maroon">L1</font> <a href="ordinal1.html#R1">c=</a>  <a href="waybel23.html#K2">weight</a> <font color="Maroon">T</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div>
