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<b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4" target="_self">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">S</font> be  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> is    <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X</font>] </span>}</span> ;<br/>
<a NAME="E1:35"/>
<span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X</font>] </span>}</span>  is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X</font>] </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E3:35"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is    <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T35">CONNSP_2:35</a></i>;<br/>
<a NAME="E2:35_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_1"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:35_1_1"/><b>then </b>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> is    <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">S</font> )
 ;<br/><b>hence </b><a NAME="E3:35_1_1"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T36">CONNSP_2:36</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:35_1"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</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><b>that </b><br/><a NAME="E5:35_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E6:35_1"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E8:35_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K3">`</a>  is    <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:35_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
;<br/>
<a NAME="E10:35_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E11:35_1"/>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 
;<br/>
<a NAME="E12:35_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<b>hence </b><a NAME="E13:35_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/>


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

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E4:35"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E5:35"/><b>then </b>
 ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> is    <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">S</font> )
 
;<br/>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">X</font> is    <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">S</font>
 ;<br/>


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