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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>thus </b><a NAME="E1:76"/>
( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  implies  ex <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">B1</font>,<font color="Olive" title="b2">B2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  &amp; <font color="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B1</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B2</font> &amp; <font color="Olive" title="b1">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1"/>
<font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">B1</font>, <font color="Maroon" title="c5">B2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:76_1"/><i><font color="Green" title="E54">A1</font></i>: 
( <font color="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B1</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> )
 <b>and </b><br/><a NAME="E4:76_1"/><i><font color="Green" title="E55">A2</font></i>: 
( <font color="Maroon" title="c4">B1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">B2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T49" target="_self" title="TSEP_1:th.49">Th49</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">B1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">B2</font>
;<br/>

<b>thus </b><a NAME="E5:76_1"/>
( <font color="Maroon" title="c4">B1</font>,<font color="Maroon" title="c5">B2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  &amp; <font color="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B1</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E3:76_1"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="tsep_1.html#E4:76_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="tsep_1.html#T54" target="_self" title="TSEP_1:th.54">Th54</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c4">B1</font>, <font color="Maroon" title="c5">B2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E3:76"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c4">B1</font>,<font color="Maroon" title="c5">B2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> 
 <b>and </b><br/><a NAME="E4:76"/><i><font color="Green" title="E55">A4</font></i>: 
( <font color="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B1</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B2</font> )
 <b>and </b><br/><a NAME="E5:76"/><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font>
 ;<br/>

<a NAME="E6:76"/><i><font color="Green" title="E57">A6</font></i>: 
( <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font> )
 
<b>by </b><i><a class="txt" href="tsep_1.html#E4:76"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E7:76"/>
( <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A2</font> )
 
<b>by </b><i><a class="txt" href="tsep_1.html#E5:76"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E8:76"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E9:76"/><b>then </b><i><font color="Green" title="E58">A7</font></i>: 
( <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="tsep_1.html#E6:76"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E10:76"/>
( <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">B1</font> )
 
<b>by </b><i><a class="txt" href="tsep_1.html#E4:76"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E11:76"/><b>then </b><i><font color="Green" title="E59">A8</font></i>: 
( <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">B1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T47" title="XBOOLE_1:th.47">XBOOLE_1:47</a></i>;<br/>
<a NAME="E12:76"/>
<font color="Maroon" title="c4">B1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">B2</font>,<font color="Maroon" title="c5">B2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">B1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<b>by </b><i><a class="txt" href="tsep_1.html#E3:76"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="tsep_1.html#D7" target="_self" title="TSEP_1:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E13:76"/>
<font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E9:76"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="tsep_1.html#E11:76"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="connsp_1.html#T8" title="CONNSP_1:th.8">CONNSP_1:8</a></i>;<br/>


</div>
