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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:42_1_1"/><i><font color="Green">E27</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_8.html#D7" target="_self">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_8.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:42_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="yellow_8.html#V2" target="_self">irreducible</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow_8.html#E1:42_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:42_1_1_1"/>
( not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D4" target="_self">YELLOW_8:def 4</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:42_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>and </b><br/><a NAME="E4:42_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>assume </b><a NAME="E5:42_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="yellow_8.html#E3:42_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_8.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E7:42_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_8.html#E4:42_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:42_1_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="yellow_8.html#V2" target="_self">irreducible</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D6" target="_self">YELLOW_8:def 6</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="yellow_8.html#R1" target="_self">is_dense_point_of</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:42_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_8.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E3:42_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E4:42_1_1_2"/><b>then </b>
not  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>hence </b><a NAME="E5:42_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="yellow_8.html#T35" target="_self">Th35</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:42_1_1"/>
(  not <font color="Maroon">c<sub>5</sub></font> <a href="yellow_8.html#R1" target="_self">is_dense_point_of</a> <font color="Maroon">c<sub>4</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="yellow_8.html#R1" target="_self">is_dense_point_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>


</div>
