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<b>let </b><font color="Maroon" title="c1">TS</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> holds <br/> ( <font color="Olive" title="b1">P</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> iff for <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>  st <font color="Olive" title="b2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b2">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> iff for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>  st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>  st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) implies <font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:77_1"/>
<font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>  st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> not <font color="Olive" title="b1">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><a NAME="E2:77_1"/><b>then </b><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D4" target="_self" title="TOPS_1:def.4">Def4</a></i>;<br/><b>let </b><font color="Maroon" title="c3">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c3">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies  not <font color="Maroon" title="c3">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><b>assume </b><b>that </b><br/><a NAME="E3:77_1"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 <b>and </b><br/><a NAME="E4:77_1"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c3">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E5:77_1"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E6:77_1"/>
<font color="Maroon" title="c3">Q</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="tops_1.html#E2:77_1"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="tops_1.html#E4:77_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="tops_1.html#E5:77_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="tops_1.html#T80" target="_self" title="TOPS_1:th.80">Th80</a></i>;<br/><b>hence </b><a NAME="E7:77_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tops_1.html#E3:77_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:77"/><i><font color="Green" title="E45">A5</font></i>: 
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>  st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a></span><br/>

<b>assume </b><a NAME="E3:77"/>
not <font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:77"/><b>then </b>
not <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#D4" target="_self" title="TOPS_1:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><b> such that </b><br/><a NAME="E6:77"/><i><font color="Green" title="E46">A6</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E7:77"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E8:77"/><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T80" target="_self" title="TOPS_1:th.80">Th80</a></i>;<br/>
<a NAME="E9:77"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E8:77"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>;<br/>
<b>hence </b><a NAME="E10:77"/>
contradiction
 <b>by </b><i><a class="txt" href="tops_1.html#E2:77"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="tops_1.html#E6:77"><i><font color="Green" title="E46">A6</font></i></a>, <a class="txt" href="tops_1.html#E7:77"><i><font color="Green" title="E47">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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