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<div class="add">

<b>let </b><font color="Maroon">Y</font> be   <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:48"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">Y</font>]
 ;<br/>

<b>let </b><font color="Maroon">S</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">y</font>;<br/>





<a NAME="E2:48"/><i><font color="Green">E47</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </span>}</span>  as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i/>;<br/>
<a NAME="E4:48"/>
( the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">S</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#T2">CANTOR_1:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> ) </span>}</span>  as    <a href="cantor_1.html#M1">Basis</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="yellow_9.html#T40">YELLOW_9:40</a></i>;<br/>
<a NAME="E6:48"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6">CONNSP_2:6</a></i>;<br/>
<a NAME="E7:48"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon">ab</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E9:48"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">ab</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ab</font> &amp; <font color="Maroon">ab</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>
<b>consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E11:48"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">ab</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">W</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">W</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E14:48"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>

<a NAME="E15:48"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<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><a NAME="E1:48_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#K1">meet</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:48_1"/>
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:48_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:48_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:48_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E7:48_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:48_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">mH</font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">H</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E17:48"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E18:48"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">mH</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</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="E1:48_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:48_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 ;<br/><a NAME="E2:48_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ab</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:48_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">a1</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">w1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:48_2_1"/><i><font color="Green">E77</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">w1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a1</font> )
 ;<br/><a NAME="E6:48_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E7:48_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">mH</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:48"/><b>then </b>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">mH</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<b>hence </b><a NAME="E20:48"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">H</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>


</div>
