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<b>let </b><font color="Maroon">GX</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X'</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">GX</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>, <font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/><b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font>, <font color="Maroon">Q1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font>
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font>
 ;<br/>

<b>assume </b><a NAME="E4:7"/><i><font color="Green">E27</font></i>: 
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a>,<a class="ref" href="connsp_1.html#D1" target="_self">CONNSP_1:def 1</a><br/>

<a NAME="E5:7"/>
( <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:7"/><b>then </b>
( <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font> &amp; <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P2</font> = <font color="Maroon">P</font>, <font color="Maroon">Q2</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font> ;<br/>
<a NAME="E8:7"/>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T47">PRE_TOPC:47</a></i>;<br/>
<b>then </b><i><font color="Green">E29</font></i>: <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q2</font></span>)</span> = 
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T47">PRE_TOPC:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E11:7"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q2</font> &amp; <font color="Maroon">P2</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q2</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E12:7"/>
<font color="Maroon">P1</font>,<font color="Maroon">Q1</font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i>, , </i>;<br/>


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