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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="compts_1.html#NV4" title="COMPTS_1:NV.4">Hausdorff</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">T</font> is  <a href="yellow_8.html#V4" title="YELLOW_8:attr.4">sober</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">irreducible</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_8.html#D6" target="_self" title="YELLOW_8:def.6">YELLOW_8:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font> &amp; ( for <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 <font color="Olive" title="b2">q</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font> 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> ) )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">d</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E2:41"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tex_2.html#D1" title="TEX_2:def.1">TEX_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">d</font> = <font color="Maroon" title="c3">d</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">d</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font> &amp; ( 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> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:41"/>
<font color="Maroon" title="c4">d</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="yellow_8.html#T28" target="_self" title="YELLOW_8:th.28">Th28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:41"/>
<font color="Maroon" title="c5">p</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font></span><br/>

<a NAME="E6:41"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_8.html#D5" target="_self" title="YELLOW_8:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:41"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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