<?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>assume </b><a NAME="E1:26"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">L1</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>let </b><font color="Maroon">B1</font> be  <a href="waybel23.html#V6">with_bottom</a>  <a href="waybel23.html#M1">CLbasis</a> of <font color="Maroon">L1</font>;<br/>

<b>consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</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">a1</font> = <font color="Maroon">a1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<a NAME="E5:26"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E6:26"/><b>then </b>
 <a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Maroon">a1</font>,<span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">L1</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Olive">a</font>,<font color="Olive">A</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Olive">A</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">WU</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Olive">a</font>,<font color="Olive">A</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Olive">A</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E8:26"/>
not <font color="Maroon">B1</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:26"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="card_4.html#T87">CARD_4:87</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">a<sub>1</sub></font>,1</span>)</span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Olive">y</font>,<font color="Olive">z</font> );<br/>
<a NAME="E10:26"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a>   ex <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">WU</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a> ;<br/>

<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> <a href="finseq_4.html#K4">/.</a> 1 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E2:26_1"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">x</font>,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T115">FINSEQ_3:115</a></i>;<br/>
<a NAME="E3:26_1"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">x</font>,1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">x</font>,1</span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:26_1"/>
 <a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Maroon">y</font>,<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="waybel31.html#K2" target="_self">Way_Up</a> <font color="Olive">a</font>,<font color="Olive">A</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Olive">A</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">u</font> =  <a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">WU</font> ;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

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

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

<b>thus </b><a NAME="E7:26_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">x</font>,1</span>)</span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">WU</font><b> such that </b><br/><a NAME="E12:26"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E13:26"/><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:26"/>
<font color="Maroon">WU</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><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="E1:26_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">WU</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font><b> such that </b><br/><a NAME="E3:26_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:26_2"/><i><font color="Green">E54</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="E5:26_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 ;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E7:26_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">B1</font> <b>by </b><i>, , <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B1</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">q</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font>;<br/>
<a NAME="E10:26_2"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">v</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font><b> such that </b><br/><a NAME="E12:26_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E13:26_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>,1</span>)</span>
 <b>and </b><br/><a NAME="E14:26_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Maroon">y</font>,<font color="Maroon">v</font>
 <b>by </b><i/>;<br/>
<a NAME="E15:26_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="finseq_5.html#T16">FINSEQ_5:16</a></i>;<br/>
<a NAME="E16:26_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 
<b>by </b><i>, , <a class="ref" href="wsierp_1.html#T48">WSIERP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E17:26_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">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>
<b>hence </b><a NAME="E15:26"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Olive">a</font>,<font color="Olive">A</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Olive">A</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> ) </span>}</span>  <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/>


</div>
