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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V6" title="PRE_TOPC:attr.6">T_0</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L1</font> being  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>  st  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L1</font> &amp; not <font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  holds <br/> <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Olive" title="b1">L1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">L1</font> be   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L1</font> &amp; not <font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  implies  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Maroon" title="c2">L1</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:12"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L1</font>
 <span class="kw">and </span><br/><a NAME="E2:12"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
not <font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Maroon" title="c2">L1</font></span><br/>

<a NAME="E3:12"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c2">L1</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span>  or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c2">L1</font> : verum </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c2">L1</font> : verum </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">B2</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E3:12_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B2</font>
 ;<br/>
<a NAME="E4:12_1"/>
<font color="Maroon" title="c4">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T78" title="TOPS_2:th.78">TOPS_2:78</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">B3</font> = <font color="Maroon" title="c4">B2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></span>;<br/>
<a NAME="E6:12_1"/>
not <font color="Maroon" title="c4">B2</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="yellow15.html#T31" title="YELLOW15:th.31">YELLOW15:31</a></span>;<br/>
<a NAME="E7:12_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A5</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c5">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow15.html#T28" title="YELLOW15:th.28">YELLOW15:28</a></span>;<br/>
<a NAME="E8:12_1"/>
 <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c5">B3</font> is   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c2">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="waybel31.html#T7" target="_self" title="WAYBEL31:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:12_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c2">L1</font> : verum </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E7:12_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:12"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c2">L1</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c2">L1</font> : verum </span>}</span>  or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_2"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c2">L1</font> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">B2</font> being   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c2">L1</font><span class="kw"> such that </span><br/><a NAME="E3:12_2"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B2</font>
 ;<br/>
<a NAME="E4:12_2"/>
<font color="Maroon" title="c4">B2</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="waybel31.html#T6" target="_self" title="WAYBEL31:th.6">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:12_2"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:12"/><span class="kw">then </span>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B1</font></span>)</span> where <font color="Olive" title="b1">B1</font> is   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c2">L1</font> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:12"/>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Maroon" title="c2">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#D5" title="WAYBEL23:def.5">WAYBEL23:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
