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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V3">Hausdorff</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>, <font color="Maroon">g</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>





<b>thus </b><a NAME="E1:55"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> </span>}</span>  holds <br/><font color="Olive">X</font> is <a href="pre_topc.html#V3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> </span>}</span> 
 ;<br/>

<a NAME="E2:55_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> <font color="Maroon">X</font> iff  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>proof </b></a><div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:55_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:55_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:55_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 <b>and </b><br/><a NAME="E6:55_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>and </b><br/><a NAME="E7:55_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a></i>;<br/><b>take </b><font color="Maroon">Q</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font></span>)</span>;<br/><a NAME="E8:55_1_1_1"/>
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><b>hence </b><a NAME="E9:55_1_1_1"/>
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/><b>thus </b><a NAME="E10:55_1_1_1"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:55_1_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:55_1_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">yf</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:55_1_1_1_1"/><i><font color="Green">E80</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">yf</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <font color="Maroon">yf</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D14">RELAT_1:def 14</a></i>;<br/>
<a NAME="E5:55_1_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">yg</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:55_1_1_1_1"/><i><font color="Green">E81</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">yg</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> &amp; <font color="Maroon">yg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D14">RELAT_1:def 14</a></i>;<br/>
<a NAME="E8:55_1_1_1_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">yf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Maroon">yg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E9:55_1_1_1_1"/>
not <font color="Maroon">yg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="txt" href="yellow12.html#E1"><i><font color="Green">Lemma30</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E10:55_1_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , <a class="txt" href="yellow12.html#E1:55"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:55_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E12:55_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E13:55_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D14">RELAT_1:def 14</a></i>;<br/><a NAME="E14:55_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E15:55_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E16:55_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D14">RELAT_1:def 14</a></i>;<br/><b>hence </b><a NAME="E17:55_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><a NAME="E2:55_1_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 ;<br/>

<b>thus </b><a NAME="E3:55_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:55_1"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57">TOPS_1:57</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">S</font>;<br/>

<b>assume </b><a NAME="E2:55"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> </span>}</span> 
 ;<br/>

<a NAME="E3:55"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><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="E1:55_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:55_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> )
 ;<br/>
<b>thus </b><a NAME="E4:55_2"/>
<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">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E5:55"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:55_3_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E2:55_3_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:55_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:55_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<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="E2:55_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:55_3"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:55_3_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:55_3_2"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> )
 ;<br/><b>thus </b><a NAME="E4:55_3_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:55_3"/><b>then </b>
<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">S</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:55_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:55"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:55"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


</div>
