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<b>let </b><font color="Maroon" title="c1">X</font> be   <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="E1:55"/>
( <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">C1</font>, <font color="Olive" title="b2">C2</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="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">C1</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">C2</font> &amp; <font color="Olive" title="b1">C1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Olive" title="b2">C2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Olive" title="b1">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c4">C1</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A1</font>;<br/>
<b>set </b><font color="Maroon" title="c5">C2</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A2</font>;<br/>
<b>assume </b><a NAME="E1:55_1"/><i><font color="Green" title="E34">A1</font></i>: 
<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>take </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A1</font>
;<br/>

<b>take </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A2</font>
;<br/>

<b>thus </b><a NAME="E2:55_1"/>
( <font color="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A2</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A2</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A1</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:55_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="connsp_1.html#D1" title="CONNSP_1:def.1">CONNSP_1:def 1</a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c4">C1</font>, <font color="Maroon" title="c5">C2</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="E3:55"/><i><font color="Green" title="E34">A2</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">C1</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">C2</font> )
 <b>and </b><br/><a NAME="E4:55"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c4">C1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon" title="c5">C2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A1</font> )
 <b>and </b><br/><a NAME="E5:55"/><i><font color="Green" title="E36">A4</font></i>: 
( <font color="Maroon" title="c4">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c5">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 ;<br/>

<a NAME="E6:55"/>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C1</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C2</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">C1</font> &amp; <font color="Maroon" title="c2">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">C2</font> )
 
<b>by </b><i><a class="txt" href="tsep_1.html#E3:55"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="tsep_1.html#E4:55"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="tsep_1.html#E5:55"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="tops_1.html#T31" title="TOPS_1:th.31">TOPS_1:31</a></i>;<br/>
<a NAME="E7:55"/><b>then </b>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon" title="c2">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<b>hence </b><a NAME="E8:55"/>
<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="ref" href="connsp_1.html#D1" title="CONNSP_1:def.1">CONNSP_1:def 1</a></i>;<br/>


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