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

<b>let </b><font color="Maroon" title="c1">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><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#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E10">A1</font></i>: 
 <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>
 ;<br/>

<b>assume </b><a NAME="E2:13"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 ;<br/>

<a NAME="E3:13"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">L1</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>
 
<b>by </b><i><a class="ref" href="yellow15.html#T26" title="YELLOW15:th.26">YELLOW15:26</a></i>;<br/>
<a NAME="E4:13"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L1</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>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E5:13"/><b>then </b><i><font color="Green" title="E12">A3</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L1</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> 
 
;<br/>
<a NAME="E6:13"/><i><font color="Green" title="E13">A4</font></i>: 
 <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Maroon" title="c2">L1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Maroon" title="c2">L1</font>
 ;<br/>

<b>hence </b><a NAME="E2:13_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L1</font>
 <b>by </b><i><a class="txt" href="waybel31.html#E5:13"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:13_2"/>
<font color="Maroon" title="c3">Z</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> 
 ;<br/><b>then consider </b><font color="Maroon" title="c4">B1</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><b> such that </b><br/><a NAME="E3:13_2"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c3">Z</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">B1</font>
 ;<br/><a NAME="E4:13_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="waybel31.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/><a NAME="E5:13_2"/><b>then </b>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="yellow15.html#T3" title="YELLOW15:th.3">YELLOW15:3</a></i>;<br/><a NAME="E6:13_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="waybel31.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><a NAME="E7:13_2"/><b>then </b><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c2">L1</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/><a NAME="E8:13_2"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B1</font>
 <b>by </b><i><a class="ref" href="waybel23.html#D8" title="WAYBEL23:def.8">WAYBEL23:def 8</a></i>;<br/><a NAME="E9:13_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B1</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T48" title="WAYBEL23:th.48">WAYBEL23:48</a></i>;<br/><a NAME="E10:13_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">B1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/><b>hence </b><a NAME="E11:13_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z</font>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:13_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="waybel31.html#E7:13_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="yellow15.html#T27" title="YELLOW15:th.27">YELLOW15:27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:13"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</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> 
 
<b>by </b><i><a class="txt" href="waybel31.html#E5:13"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>;<br/>
<b>hence </b><a NAME="E9:13"/>
 <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Maroon" title="c2">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L1</font>
 <b>by </b><i><a class="txt" href="waybel31.html#E6:13"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
