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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K5" target="_self">ClFinTop</a> <font color="Maroon">X</font></span>)</span>;<br/>



<a NAME="E1:49"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K5" target="_self">ClFinTop</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="topgen_3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:49"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K5" target="_self">ClFinTop</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  &amp;  <a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K5" target="_self">ClFinTop</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
;<br/>
<a NAME="E3:49"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="topgen_3.html#K5" target="_self">ClFinTop</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  )
 ;<br/><a NAME="E2:49_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E3:49_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K5" target="_self">ClFinTop</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="topgen_3.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/><b>hence </b><a NAME="E4:49_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:49"><i><font color="Green">E38</font></i></a>, <a class="ref" href="topgen_3.html#T1" target="_self">Th1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:49"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="finset_1.html#V1">finite</a> )
 ;<br/>

<a NAME="E6:49"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="topgen_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E7:49"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>


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