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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">sober</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<a NAME="E1:34"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green" title="E24">A1</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:34_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">irreducible</a>
 
<b>by </b><i><a class="ref" href="yellow_8.html#T26" target="_self" title="YELLOW_8:th.26">Th26</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">r</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:34_1"/>
<font color="Maroon" title="c4">r</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E5:34_1"/><i><font color="Green" title="E25">A2</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">q</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#D6" target="_self" title="YELLOW_8:def.6">Def6</a></i>;<br/>
<a NAME="E6:34_1"/>
( <font color="Maroon" title="c2">p</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c3">q</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="yellow_8.html#T27" target="_self" title="YELLOW_8:th.27">Th27</a></i>;<br/>
<a NAME="E7:34_1"/><b>then </b>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> )
 
<b>by </b><i><a class="txt" href="yellow_8.html#E1:34_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="yellow_8.html#E5:34_1"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:34_1"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34"/>
<font color="Maroon" title="c1">T</font> is <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">T_0</a>
 <b>by </b><i><a class="ref" href="yellow_8.html#T32" target="_self" title="YELLOW_8:th.32">Th32</a></i>;<br/>


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