<?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="t_0topsp.html#V2">T_0</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#NM7">sup-Semilattice</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:10"/><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>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">T</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<a NAME="E3:10"/><i><font color="Green">E43</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">B1</font></span>)</span> where <font color="Olive">B1</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">B1</font></span>)</span> where <font color="Olive">B1</font> is  <a href="waybel23.html#V6">with_bottom</a>  <a href="waybel23.html#M1">CLbasis</a> of <font color="Maroon">L1</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:10_1"/>
<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="card_1.html#K1">Card</a> <font color="Olive">B1</font></span>)</span> where <font color="Olive">B1</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">B2</font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B2</font>
 ;<br/>
<a NAME="E4:10_1"/>
<font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B3</font> = <font color="Maroon">B2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> <b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E6:10_1"/>
not <font color="Maroon">B2</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="E7:10_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B2</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">B3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow15.html#T28">YELLOW15:28</a></i>;<br/>
<a NAME="E8:10_1"/>
 <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">B3</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>, </i>;<br/>
<b>hence </b><a NAME="E9:10_1"/>
<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="card_1.html#K1">Card</a> <font color="Olive">B1</font></span>)</span> where <font color="Olive">B1</font> is  <a href="waybel23.html#V6">with_bottom</a>  <a href="waybel23.html#M1">CLbasis</a> of <font color="Maroon">L1</font> : verum </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:10"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">B1</font></span>)</span> where <font color="Olive">B1</font> is  <a href="waybel23.html#V6">with_bottom</a>  <a href="waybel23.html#M1">CLbasis</a> of <font color="Maroon">L1</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">B1</font></span>)</span> where <font color="Olive">B1</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:10_2"/>
<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="card_1.html#K1">Card</a> <font color="Olive">B1</font></span>)</span> where <font color="Olive">B1</font> is  <a href="waybel23.html#V6">with_bottom</a>  <a href="waybel23.html#M1">CLbasis</a> of <font color="Maroon">L1</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">B2</font> being  <a href="waybel23.html#V6">with_bottom</a>  <a href="waybel23.html#M1">CLbasis</a> of <font color="Maroon">L1</font><b> such that </b><br/><a NAME="E3:10_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B2</font>
 ;<br/>
<a NAME="E4:10_2"/>
<font color="Maroon">B2</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:10_2"/>
<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="card_1.html#K1">Card</a> <font color="Olive">B1</font></span>)</span> where <font color="Olive">B1</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font> : verum </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:10"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">B1</font></span>)</span> where <font color="Olive">B1</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">B1</font></span>)</span> where <font color="Olive">B1</font> is  <a href="waybel23.html#V6">with_bottom</a>  <a href="waybel23.html#M1">CLbasis</a> of <font color="Maroon">L1</font> : verum </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:10"/>
 <a href="waybel23.html#K2">weight</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="waybel31.html#K1" target="_self">CLweight</a> <font color="Maroon">L1</font>
 <b>by </b><i><a class="ref" href="waybel23.html#D5">WAYBEL23:def 5</a></i>;<br/>


</div>
