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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:40"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><br/> for <font color="Olive">U</font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> holds <br/> ex <font color="Olive">W</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <font color="Olive">W</font> &amp;  <a href="prob_1.html#K2">Union</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> (  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Olive">W</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> &amp; <font color="Olive">W</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font> is <a href="pre_topc.html#V3">open</a> ) ) )
 ;<br/>

<a NAME="E2:40"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">B</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> holds <br/> ex <font color="Olive">UA</font>, <font color="Olive">WB</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">UA</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">WB</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">UA</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">WB</font> &amp; <font color="Olive">UA</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">WB</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/>

<b>set </b><font color="Maroon">U</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>;<br/>
<a NAME="E2:40_1"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E3:40_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E4:40_1"/><i><font color="Green">E60</font></i>: 
( <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>consider </b><font color="Maroon">Un</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:40_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">Un</font> &amp;  <a href="prob_1.html#K2">Union</a> <font color="Maroon">Un</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> (  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font> is <a href="pre_topc.html#V3">open</a> ) ) )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">Wn</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:40_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">Wn</font> &amp;  <a href="prob_1.html#K2">Union</a> <font color="Maroon">Wn</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> (  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> &amp; <font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font> is <a href="pre_topc.html#V3">open</a> ) ) )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E9:40_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:40_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</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="E2:40_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:40_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:40_1_2"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:40_1_2"/>
<font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> = <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> </span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/>
<a NAME="E11:40_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">FWU</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E13:40_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E14:40_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">CLn</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> ;<br/>
<a NAME="E1:40_1_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_3_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:40_1_3_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><b>thus </b><a NAME="E4:40_1_3_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_1_3_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">CLn</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:40_1_3"/><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> =  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E4:40_1_3"/><i><font color="Green">E110</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i>
<a NAME="E5:40_1_3"/>
<font color="Maroon">CLn</font> is <a href="tops_2.html#V2">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:40_1_3_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CLn</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:40_1_3_2_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><b>thus </b><a NAME="E4:40_1_3_2_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:40"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_1_3_2"/>
<font color="Maroon">CLn</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/>


</div><b>end;</b></div>
<a NAME="E6:40_1_3"/><b>then </b>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">CLn</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T28">TOPS_2:28</a></i>;<br/>
<a NAME="E7:40_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</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">CLn</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:40_1_3"/>
<font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> = <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> </span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/>
<a NAME="E15:40_1"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">FUW</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E17:40_1"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E18:40_1"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">CLn</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> ;<br/>
<a NAME="E1:40_1_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_4_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:40_1_4_1_1"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><b>thus </b><a NAME="E4:40_1_4_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma34</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_1_4_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">CLn</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">k</font></span>)</span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:40_1_4"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> =  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E4:40_1_4"/><i><font color="Green">E119</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma34</font></i></a>);<br/></i>
<a NAME="E5:40_1_4"/>
<font color="Maroon">CLn</font> is <a href="tops_2.html#V2">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:40_1_4_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CLn</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:40_1_4_2_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><b>thus </b><a NAME="E4:40_1_4_2_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_1_4_2"/>
<font color="Maroon">CLn</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/>


</div><b>end;</b></div>
<a NAME="E6:40_1_4"/><b>then </b>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">CLn</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T28">TOPS_2:28</a></i>;<br/>
<a NAME="E7:40_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</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">CLn</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:40_1_4"/>
<font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E19:40_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FWU</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_5_1"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FWU</font> )
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:40_1_5_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/><a NAME="E4:40_1_5_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_5_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">CL</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_5_1_1"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CL</font> &amp; <font color="Maroon">CL</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:40_1_5_1_1"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">CL</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E6:40_1_5_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E7:40_1_5_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E5:40_1_5_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>)
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:40_1_5_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/><b>hence </b><a NAME="E7:40_1_5_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_1_5"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FWU</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:40_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FUW</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_6_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="E1:40_1_6_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FUW</font> )
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:40_1_6_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/><a NAME="E4:40_1_6_1"/>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_6_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_6_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">CL</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_6_1_1"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CL</font> &amp; <font color="Maroon">CL</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:40_1_6_1_1"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">CL</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E6:40_1_6_1_1"/>
<font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E7:40_1_6_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E5:40_1_6_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">k</font>)
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:40_1_6_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:40_1_6_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_1_6"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FUW</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:40_1"/><i><font color="Green">E165</font></i>: 
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">FUW</font> <a href="xboole_0.html#R1">misses</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FWU</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_7"/>
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">FUW</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FWU</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_7"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FUW</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FWU</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:40_1_7"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:40_1_7"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<a NAME="E8:40_1_7"/><i><font color="Green">E169</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_7_1"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_7_1"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:40_1_7_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">l</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> </span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a>, </i>;<br/>
<a NAME="E5:40_1_7_1"/><b>then </b><i><font color="Green">E173</font></i>: 
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">l</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:40_1_7_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">l</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:40_1_7_1"/><b>then </b><i><font color="Green">E174</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:40_1_7_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R2">in</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#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E9:40_1_7_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> or ( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> ) )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:40_1_7_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:40_1_7_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">l</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E12:40_1_7_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:40_1_7"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_7_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_7_2"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_7_2"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FUW</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FWU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:40_1_7_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">l</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:40_1_7_2"/><b>then </b><i><font color="Green">E178</font></i>: 
not <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">l</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:40_1_7_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><font color="Maroon">Un</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">l</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> </span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a>, </i>;<br/>
<a NAME="E7:40_1_7_2"/><b>then </b><i><font color="Green">E179</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:40_1_7_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E9:40_1_7_2"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> or ( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> &amp;  <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#K2">Seg</a> <font color="Maroon">n</font> ) )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:40_1_7_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:40_1_7_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">l</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E12:40_1_7_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:40_1_7"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nagata_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:40_1"/>
(  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FUW</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="prob_1.html#K2">Union</a> <font color="Maroon">FWU</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="txt" href="nagata_1.html#E1:40"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E23:40_1"/>
 ex <font color="Olive">UA</font>, <font color="Olive">WB</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">UA</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">WB</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">UA</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">WB</font> &amp; <font color="Olive">UA</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">WB</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma34</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
<font color="Maroon">T</font> <a href="compts_1.html#NR4" target="_self">is_T4</a> 
 <b>by </b><i><a class="ref" href="compts_1.html#D6">COMPTS_1:def 6</a></i>;<br/>


</div>
