<?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">W</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">W</font> is <a href="tops_2.html#V1" target="_self">open</a>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">W</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:21"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:21"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">Z</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font>  st <font color="Olive">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/> <a href="setfam_1.html#K6">meet</a> <font color="Olive">Z</font> is <a href="pre_topc.html#V3">open</a>;<br/>
<a NAME="E7:21"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:21_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span>
 <b>and </b><br/><a NAME="E2:21_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:21_1"/><i><font color="Green">E45</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T149">RELAT_1:149</a>, <a class="ref" href="setfam_1.html#T2">SETFAM_1:2</a></i>;<br/>
<a NAME="E4:21_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">GX</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a></i>;<br/>
<b>hence </b><a NAME="E5:21_1"/>
(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Z</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:21"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">Z</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font>  st <font color="Olive">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/> <a href="setfam_1.html#K6">meet</a> <font color="Olive">Z</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

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

<b>assume </b><a NAME="E2:21_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E3:21_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>

<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:21_2_1"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/><a NAME="E3:21_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/><a NAME="E4:21_2_1"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>;<br/><a NAME="E5:21_2_1"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:21_2_1"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Z</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2_2"/><i><font color="Green">E53</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:21_2_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E3:21_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:21_2_2"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:21_2_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E6:21_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:21_2_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><i><font color="Green">E64</font></i>: <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T11">FINSEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T153">RELAT_1:153</a></i>
;<br/><a NAME="E9:21_2_2"/>
<font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/><b>then reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font> <b>by </b><i/>;<br/><a NAME="E11:21_2_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">G1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/><a NAME="E12:21_2_2"/>
<font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/><b>then reconsider </b><font color="Maroon">G2</font> = <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font> <b>by </b><i/>;<br/><a NAME="E14:21_2_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E15:21_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E16:21_2_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E17:21_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E18:21_2_2"/><b>then </b>
<font color="Maroon">G2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/><a NAME="E19:21_2_2"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">G2</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>;<br/><a NAME="E20:21_2_2"/>
<font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/><a NAME="E21:21_2_2"/><b>then </b><i><font color="Green">E70</font></i>: 
(  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>;<br/><a NAME="E22:21_2_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E23:21_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E24:21_2_2"/><b>then </b>
(  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">G2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E25:21_2_2"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Z</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:21_2"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Z</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:21"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_3"/><i><font color="Green">E73</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:21_3"/>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/><b>hence </b><a NAME="E3:21_3"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:21_4"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/><a NAME="E3:21_4"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a>, <a class="ref" href="setfam_1.html#T2">SETFAM_1:2</a></i>;<br/><a NAME="E4:21_4"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">GX</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a></i>;<br/><b>hence </b><a NAME="E5:21_4"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_5"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:21_5"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><b>hence </b><a NAME="E3:21_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:21"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/>


</div>
