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<div class="add">

<b>set </b><font color="Maroon">T</font> =  <a href="yellow_8.html#K1" target="_self">CofinTop</a> <font color="Maroon">X</font>;<br/>
<a NAME="E1:42_1_1"/><i><font color="Green">E19</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">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">S</font> =  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</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">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E3:42_1_1"/>
<font color="Maroon">S</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">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</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">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow_8.html#E4:42_1_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">S</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">S</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">S1</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">X</font></span>)</span>, <font color="Maroon">S2</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">X</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:42_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">S1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">S2</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">E27</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">S2</font>
 ;<br/>

<b>assume </b><a NAME="E5:42_1_1_1"/>
( <font color="Maroon">S1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">S</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">S1'</font> = <font color="Maroon">S1</font>, <font color="Maroon">S2'</font> = <font color="Maroon">S2</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E7:42_1_1_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1'</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S2'</font>
 
<b>by </b><i><a class="ref" href="yellow_8.html#T4" target="_self">Th4</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">S</font> = <font color="Maroon">S</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">X</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">S</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">p</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">X</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">p</font> <a href="yellow_8.html#R1" target="_self">is_dense_point_of</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:42_1_1_2"/><b>then </b>
<font color="Maroon">S</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">p</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:42_1_1_2"/><b>then </b>
<font color="Maroon">X</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">p</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">p</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/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:42_1_1"/>
(  not <font color="Maroon">p</font> <a href="yellow_8.html#R1" target="_self">is_dense_point_of</a> <font color="Maroon">S</font> or  ex <font color="Olive">q</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">X</font></span>)</span> st <br/>( <font color="Olive">q</font> <a href="yellow_8.html#R1" target="_self">is_dense_point_of</a> <font color="Maroon">S</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> ) )
 ;<br/>


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