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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:52_1_1"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="openlatt.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E2:52_1_1"/><i><font color="Green">E43</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="openlatt.html#D12" target="_self">Def12</a></i>;<br/>
<b>thus </b><a NAME="E3:52_1_1"/>
not  <a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_1_1_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="openlatt.html#E1:52_1_1"><i><font color="Green">E42</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a><br/>


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

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="openlatt.html#K16" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E5:52_1_1"/>
 <a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K16" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="openlatt.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E6:52_1_1"/><b>then </b>
 <a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>
<b>hence </b><a NAME="E7:52_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="openlatt.html#E1:52_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="openlatt.html#E2:52_1_1"><i><font color="Green">E43</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means  <a href="tarski.html#K3">union</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/><b>assume </b><a NAME="E1:52_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  </span>}</span> ;<br/><a NAME="E2:52_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1_1_2_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:52_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="openlatt.html#E1:52_1_1_2"><i><font color="Green">E44</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:52_1_1_2_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E2:52_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E5:52_1_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="openlatt.html#E1:52_1_1_2_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="openlatt.html#E4:52_1_1_2_1_1"><i><font color="Green">E47</font></i></a></i>;<br/><b>hence </b><a NAME="E6:52_1_1_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  </span>}</span> 
 <b>by </b><i><a class="txt" href="openlatt.html#E4:52_1_1_2_1_1"><i><font color="Green">E47</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>7</sub></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="E2:52_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:52_1_1_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  )
 ;<br/>
<a NAME="E5:52_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="openlatt.html#E4:52_1_1_2_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:52_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E4:52_1_1_2_1"><i><font color="Green">E46</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i><a NAME="E4:52_1_1_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E2:52_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="borsuk_1.html#T17">BORSUK_1:17</a></i>;<br/><b>hence </b><a NAME="E5:52_1_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="openlatt.html#E2:52_1_1"><i><font color="Green">E43</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E9:52_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E11:52_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E2:52_1_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E13:52_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E2:52_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="openlatt.html#E9:52_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E14:52_1_1"/>
 <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></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="E1:52_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1_1_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:52_1_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K16" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E3:52_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="openlatt.html#T15" target="_self">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:52_1_1_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K16" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="openlatt.html#E3:52_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="openlatt.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E8:52_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K16" target="_self">StoneH</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="openlatt.html#E3:52_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="openlatt.html#E5:52_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="openlatt.html#E7:52_1_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="openlatt.html#T17" target="_self">Th17</a></i>;<br/>
<b>hence </b><a NAME="E9:52_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E16:52_1_1"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T39">SETFAM_1:39</a></i>;<br/>
<b>hence </b><a NAME="E17:52_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K17" target="_self">HTopSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="openlatt.html#E2:52_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="openlatt.html#E11:52_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="openlatt.html#E13:52_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>


</div>
