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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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">A</font> <a href="tsep_2.html#R1" target="_self">constitute_a_decomposition</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1"/>
(  not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><a NAME="E1:8_1_1"/><i><font color="Green">E13</font></i>: 
not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:8_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:8_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><a NAME="E4:8_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</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/><b>hence </b><a NAME="E5:8_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tsep_2.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><a NAME="E1:8_1_2"/>
<font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then </b> <a href="xboole_0.html#K1">{}</a>  = 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font>

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:8_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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