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<b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E2:38"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:38_1"/><b>then </b>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/><b>hence </b><a NAME="E3:38_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:38"/><b>then </b>
<font color="Maroon">F</font> is <a href="tops_2.html#V2">closed</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:38"/>
( <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) </span>}</span>  implies <font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T28">TOPS_2:28</a></i>;<br/>


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