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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">X1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">X2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>





<b>reconsider </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>, <font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>thus </b><a NAME="E2:31"/>
( <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_2.html#R3" target="_self">constitute_a_decomposition</a>  implies ( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font> &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
for <font color="Olive">A1</font>, <font color="Olive">A2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> holds <br/><font color="Olive">A1</font>,<font color="Olive">A2</font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tsep_2.html#D2" target="_self">TSEP_2:def 2</a><br/>

<a NAME="E2:31_1"/><b>then </b><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/>
<a NAME="E3:31_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:31_1"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>

<a NAME="E5:31_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tsep_2.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E6:31_1"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a>, <a class="ref" href="tsep_1.html#D2">TSEP_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T5">TSEP_1:5</a></i>;<br/>


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

<b>assume </b><a NAME="E3:31"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font>
 ;<br/>

<b>assume </b><a NAME="E4:31"/><i><font color="Green">E26</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">A2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:31_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> )
 ;<br/><a NAME="E2:31_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_2.html#T12" target="_self">Th12</a>, <a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/><a NAME="E3:31_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tsep_2.html#T14" target="_self">Th14</a>, <a class="txt" href="tsep_2.html#E4:31"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tsep_1.html#D2">TSEP_1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:31_2"/>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 <b>by </b><i><a class="txt" href="tsep_2.html#E1:31_2"><i><font color="Green">E27</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_2.html#R3" target="_self">constitute_a_decomposition</a> 
 <b>by </b><i><a class="ref" href="tsep_2.html#T9" target="_self">Th9</a></i>;<br/>


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