<?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>set </b><font color="Maroon">L</font> =  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E1:73"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:73"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">T</font> <a href="compts_1.html#NR3" target="_self">is_T3</a>  &amp;  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> is <a href="waybel_3.html#V3" target="_self">continuous</a> )
 ;<br/>

<a NAME="E3:73"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> is  <a href="waybel_3.html#V3" target="_self">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D9" target="_self">WAYBEL_3:def 9</a><br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E4:73"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<font color="Maroon">a</font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_3.html#E1:73"><i><font color="Green">E34</font></i></a>, </i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>
;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:73"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:73"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E9:73"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T2" target="_self">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="waybel_3.html#D2" target="_self">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span><b> such that </b><br/><a NAME="E12:73"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="waybel_3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E13:73"/>
<font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="waybel_3.html#T2" target="_self">Th2</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E15:73"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="waybel_3.html#T1" target="_self">Th1</a>, <a class="ref" href="waybel_3.html#T2" target="_self">Th2</a>, <a class="ref" href="urysohn1.html#T29">URYSOHN1:29</a></i>;<br/>
<b>take </b><font color="Maroon">Z</font> =  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">W</font>;<br/>

<a NAME="E16:73"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E17:73"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>

<a NAME="E18:73"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:73"/><b>then </b>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E20:73"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D7">COMPTS_1:def 7</a><br/>

<b>assume </b><a NAME="E21:73"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">F</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">ZZ</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">ZZ</font> = <font color="Maroon">ZZ</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">FZ</font> = <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">ZZ</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E25:73"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font> holds <br/><font color="Olive">x</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E26:73"/><b>then </b>
<font color="Maroon">ZZ</font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E27:73"/><b>then </b>
<font color="Maroon">FZ</font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E3:73"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tops_2.html#T20">TOPS_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">FF</font> = <font color="Maroon">FZ</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T25">YELLOW_1:25</a></i>;<br/>
<i><font color="Green">E49</font></i>:  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">FF</font> = 
 <a href="tarski.html#K3">union</a> <font color="Maroon">FZ</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">ZZ</font></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
;<br/>
<a NAME="E30:73"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E3:73"><i><font color="Green">E39</font></i></a>, <a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<a NAME="E31:73"/><b>then </b>
<font color="Maroon">Z</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">FF</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E4:73"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E32:73"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">FF</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>;<br/>
<a NAME="E33:73"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">FF</font>
 
;<br/>
<a NAME="E34:73"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">FF</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E35:73"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">FF</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span><b> such that </b><br/><a NAME="E37:73"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FF</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E38:73"/><i><font color="Green">E51</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>;<br/>
<a NAME="E39:73"/>
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">ZZ</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">ZZ</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E41:73"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/>

<b>thus </b><a NAME="E42:73"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">G</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:73_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:73_3"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, , <a class="ref" href="waybel_3.html#T3" target="_self">Th3</a>, <a class="txt" href="waybel_3.html#E8:73"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:73_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:73_3"/>
not <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E12:73"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_3.html#E15:73"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:73_3"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E15:73"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:73_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E15:73"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E43:73"/>
<font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>


</div>
