<?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">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:37"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="funct_1.html#NM1">Function</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">L</font>;<br/>
<a NAME="E3:37"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">p</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#M2">Element</a> of  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:37_1"/><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> is <a href="xboole_0.html#V1">empty</a> or  not  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T42">YELLOW_0:42</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_2"/>
not  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1_2"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T54">YELLOW_0:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:37_1"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">L</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">L</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">B1</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">L</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">u</font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">L</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B1</font> <b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>thus </b><a NAME="E5:37_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">u</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> , <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B1</font><b> such that </b><br/><a NAME="E5:37"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">p</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">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E6:37"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_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="E1:37_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</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">L</font> ;<br/>
<a NAME="E3:37_2"/><i><font color="Green">E37</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">z1</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T38">YELLOW_0:38</a></i>;<br/>
<a NAME="E4:37_2"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z1</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</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="E1:37_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z1</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:37_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:37_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z1</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<a NAME="E6:37_2"/><b>then </b>
<font color="Maroon">z1</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">L</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">B</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">L</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:37_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B</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>
<a NAME="E7:37"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <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">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E8:37"/><i><font color="Green">E53</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="E9:37"/>
 <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B</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="37_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="E1:37_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:37_3"/><b>then </b>
<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#K1">"\/"</a> <font color="Olive">Y</font>,<font color="Maroon">L</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">B</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">L</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">Y</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E4:37_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E5:37_3"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Y</font>,<font color="Maroon">L</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="E7:37_3"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</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">p1</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of <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/>
<a NAME="E10:37_3"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p1</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:37_3"/>
<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>
<a NAME="E10:37"/><b>then </b>
 <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">B1</font></span>)</span> <a href="tarski.html#R1">c=</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_1.html#T28">CARD_1:28</a></i>;<br/>
<a NAME="E11:37"/><b>then </b>
 <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">B1</font></span>)</span> <a href="tarski.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_4.html#T87">CARD_4:87</a></i>;<br/>
<b>hence </b><a NAME="E12:37"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <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">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
