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

<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">x'</font> = <font color="Maroon" title="c2">x</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">InIdS</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">S</font></span>)</span> as   <a href="waybel_5.html#NM3" title="WAYBEL_5:NM.3">CLSubFrame</a> of  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="waybel13.html#T8" target="_self" title="WAYBEL13:th.8">Th8</a></i>;<br/>
<b>thus </b><a NAME="E3:12"/>
( <font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> implies <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">S</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 ;<br/>

<a NAME="E2:12_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">InIdS</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D1" title="WAYBEL_8:def.1">WAYBEL_8:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">F</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span>)</span><b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c5">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E5:12_1"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font> : <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> </span>}</span> 
 <b>and </b><br/><a NAME="E6:12_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="waybel13.html#T7" target="_self" title="WAYBEL13:th.7">Th7</a></i>;<br/>
<a NAME="E7:12_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/>
<a NAME="E8:12_1"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x'</font> &amp; <font color="Olive" title="b1">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">x'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c6">F'</font> = <font color="Maroon" title="c5">F</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">F'</font> = <font color="Maroon" title="c6">F'</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">y</font> =  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">F'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c8">y</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_1_1"/>
( <font color="Maroon" title="c7">F'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1_1_1"/>
<font color="Maroon" title="c7">F'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_1_1_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x'</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:12_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="waybel13.html#E6:12_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="waybel_0.html#T42" title="WAYBEL_0:th.42">WAYBEL_0:42</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_1_1_2"/>
<font color="Maroon" title="c7">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:12_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D11" title="YELLOW_0:def.11">YELLOW_0:def 11</a></i>;<br/><b>hence </b><a NAME="E3:12_1_1_1_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x'</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#T21" title="WAYBEL_4:th.21">WAYBEL_4:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:12_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x'</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><b>assume </b><a NAME="E1:12_1_1_2"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x'</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1_1_2_1"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">u'</font> = <font color="Maroon" title="c10">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="waybel13.html#E7:12_1"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/><a NAME="E3:12_1_1_2_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c7">F'</font>,<font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:12_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="waybel13.html#E1:12_1_1_2_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="yellow_0.html#T54" title="YELLOW_0:th.54">YELLOW_0:54</a></i>;<br/><a NAME="E4:12_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/><a NAME="E5:12_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c11">u'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:12_1_1_2_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/><b>hence </b><a NAME="E6:12_1_1_2_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/></div><b>end;</b></div><a NAME="E3:12_1_1_2"/><b>then </b><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E4:12_1_1_2"/>
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c8">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/><a NAME="E5:12_1_1_2"/><b>then </b>
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font> : <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel13.html#E3:12_1_1_2"><i><font color="Green" title="E17">A7</font></i></a></i>;<br/><a NAME="E6:12_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:12_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="waybel13.html#E1:12_1_1_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E7:12_1_1_2"/>
<font color="Maroon" title="c9">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">x'</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:12_1"/>
<font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D21" title="WAYBEL_0:def.21">WAYBEL_0:def 21</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:12"/>
( <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">S</font> implies <font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green" title="E12">A8</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x'</font>
 <b>and </b><br/><a NAME="E4:12_2"/><i><font color="Green" title="E13">A9</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">x'</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D21" title="WAYBEL_0:def.21">WAYBEL_0:def 21</a></i>;<br/>
<a NAME="E5:12_2"/>
 ex <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">Y</font> where <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font> : <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c6">F</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">F</font>
;<br/>

<b>thus </b><a NAME="E2:12_2_1"/>
<font color="Maroon" title="c6">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<a NAME="E3:12_2_1"/>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E3:12_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:12_2_1"/><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>

<a NAME="E5:12_2_1"/>
<font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
;<br/>
<a NAME="E6:12_2_1"/><b>then </b><i><font color="Green" title="E15">A11</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">S</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="E7:12_2_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">S</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/>
<a NAME="E8:12_2_1"/><b>then </b><i><font color="Green" title="E16">A12</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font> : <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel13.html#E6:12_2_1"><i><font color="Green" title="E15">A11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:12_2_1_1"/>
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> implies for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">Y</font> where <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font> : <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> </span>}</span>  holds <br/><font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_2_1_1_1"/><i><font color="Green" title="E17">A13</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">z'</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="yellow_2.html#T44" title="YELLOW_2:th.44">YELLOW_2:44</a></i>;<br/>
<b>let </b><font color="Maroon" title="c9">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E3:12_2_1_1_1"/>
<font color="Maroon" title="c9">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font> : <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">Z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font><b> such that </b><br/><a NAME="E5:12_2_1_1_1"/><i><font color="Green" title="E18">A14</font></i>: 
<font color="Maroon" title="c10">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Z</font>
 <b>and </b><br/><a NAME="E6:12_2_1_1_1"/><i><font color="Green" title="E19">A15</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Z1</font>
 ;<br/>
<a NAME="E7:12_2_1_1_1"/><i><font color="Green" title="E20">A16</font></i>: 
<font color="Maroon" title="c10">Z1</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/>
<a NAME="E8:12_2_1_1_1"/><i><font color="Green" title="E21">A17</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:12_2_1_1_1"/>
<font color="Maroon" title="c8">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E4:12_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="waybel13.html#E1:12_2_1_1_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E10:12_2_1_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:12_2_1_1_1"><i><font color="Green" title="E18">A14</font></i></a>, <a class="txt" href="waybel13.html#E6:12_2_1_1_1"><i><font color="Green" title="E19">A15</font></i></a>, <a class="txt" href="waybel13.html#E7:12_2_1_1_1"><i><font color="Green" title="E20">A16</font></i></a>, <a class="txt" href="waybel13.html#E8:12_2_1_1_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:12_2_1_1"/>
( ( for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">Y</font> where <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font> : <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> </span>}</span>  holds <br/><font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_2_1_1_2"/><i><font color="Green" title="E17">A18</font></i>: 
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">Y</font> where <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font> : <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> </span>}</span>  holds <br/><font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font>
 ;<br/>

<a NAME="E2:12_2_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font> : <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel13.html#E4:12_2_1"><i><font color="Green" title="E14">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:12_2_1_1_2"><i><font color="Green" title="E17">A18</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:12_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">InIdS</font> : <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel13.html#E8:12_2_1"><i><font color="Green" title="E16">A12</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>
<a NAME="E6:12_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">InIdS</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel13.html#T7" target="_self" title="WAYBEL13:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E7:12_2"/>
<font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" title="WAYBEL_8:def.1">WAYBEL_8:def 1</a></i>;<br/>


</div><b>end;</b></div>


</div>
