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

<b>let </b><font color="Maroon">GX</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font>;<br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">A</font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>;<br/><b>consider </b><font color="Maroon">R1</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> ;<br/><a NAME="E3:39_1_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 <b>by </b><i/>;<br/><a NAME="E4:39_1_1_1"/>
( <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> iff <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/><a NAME="E5:39_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B1</font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <b>by </b><i>, </i>;<br/><a NAME="E6:39_1_1_1"/><b>then </b>
<font color="Maroon">B1</font> is <a href="pre_topc.html#V3" target="_self">open</a>
 <b>by </b><i><a class="txt" href="pre_topc.html#E1:39"><i><font color="Green">E19</font></i></a>, </i>;<br/><b>hence </b><a NAME="E7:39_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">GX</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E5:39_1_1"/><b>then </b>
<font color="Maroon">R1</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">GX</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E6:39_1_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">GX</font>
 <b>by </b><i/>;<br/><a NAME="E7:39_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font> is <a href="pre_topc.html#V3" target="_self">open</a>
 <b>by </b><i/>;<br/><a NAME="E8:39_1_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font>
 <b>by </b><i/>;<br/><a NAME="E9:39_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font></span>)</span> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <b>by </b><i>, </i>;<br/><a NAME="E10:39_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GX</font> holds <br/>( ( for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) iff for <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_2"/><i><font color="Green">E44</font></i>: 
<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">GX</font>
 ;<br/>

<b>thus </b><a NAME="E2:39_1_1_2"/>
( ( for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) implies for <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> holds <br/> not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1_2_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:39_1_1_2_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font>
 ;<br/>

<a NAME="E3:39_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:39_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:39_1_1_2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>assume </b><a NAME="E3:39_1_1_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> holds <br/> not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font>
 ;<br/>

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

<b>assume </b><a NAME="E4:39_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E5:39_1_1_2"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:39_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:39_1_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:39_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="pre_topc.html#E3:39_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:39_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font></span>)</span> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:39_1_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font></span>)</span> implies <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1_3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font></span>)</span>
 ;<br/>

<a NAME="E2:39_1_1_3_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:39_1_1_3_1"/><b>then </b>
for <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> holds <br/> not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:39_1_1_3_1"/><b>then </b>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E1:39_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pre_topc.html#E3:39_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:39_1_1_3_1"/>
<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>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:39_1_1_3"/><i><font color="Green">E50</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>
 ;<br/>

<a NAME="E3:39_1_1_3"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E3:39_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E4:39_1_1_3"/><b>then </b>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> holds <br/> not <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E1:39_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E5:39_1_1_3"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:39_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">R1</font></span>)</span>
 <b>by </b><i><a class="txt" href="pre_topc.html#E3:39_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:39_1_1"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <b>by </b><i><a class="txt" href="pre_topc.html#E1:39"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:39_1_2"/>
 <a href="pre_topc.html#K1" target="_self">{}</a> <font color="Maroon">GX</font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GX</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">GX</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:39_1_2_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font> is <a href="pre_topc.html#V3" target="_self">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:39_1_2_1"/>
 <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1" target="_self">{}</a> <font color="Maroon">GX</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:39_1_2_1"/>
 <a href="pre_topc.html#K1" target="_self">{}</a> <font color="Maroon">GX</font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:39_1_2"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
