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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">B1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">B2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>









<b>assume </b><a NAME="E1:14"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 ;<br/>

<b>assume </b><a NAME="E2:14"/>
<font color="Maroon">B1</font>,<font color="Maroon">B2</font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 ;<br/>

<a NAME="E3:14"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">B1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:14"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E5:14"/>
<font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="txt" href="tsep_2.html#E1:14"><i><font color="Green">E12</font></i></a>, </i>;<br/>
<a NAME="E6:14"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font></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"><font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="tsep_2.html#D1" target="_self">Def1</a>, </i>
;<br/>
<a NAME="E8:14"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T24">XBOOLE_1:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">B2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<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#K4">\/</a> <font color="Maroon">B2</font></span>)</span> <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#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
<b>by </b><i><a class="txt" href="tsep_2.html#E1:14"><i><font color="Green">E12</font></i></a>, <a class="txt" href="tsep_2.html#E3:14"><i><font color="Green">E13</font></i></a>, </i>
<br/>.= 
<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#K4">\/</a> <font color="Maroon">B2</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="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<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="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="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>

;<br/>
<b>hence </b><a NAME="E10:14"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font>,<font color="Maroon">A2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 <b>by </b><i>, </i>;<br/>


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