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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>, <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>, <font color="Maroon">Y1</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>, <font color="Maroon">Y2</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>reconsider </b><font color="Maroon">B1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y1</font>, <font color="Maroon">B2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y2</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>assume </b><a NAME="E3:35"/>
( <font color="Maroon">X1</font>,<font color="Maroon">Y1</font> <a href="tsep_2.html#R4" target="_self">constitute_a_decomposition</a>  &amp; <font color="Maroon">X2</font>,<font color="Maroon">Y2</font> <a href="tsep_2.html#R4" target="_self">constitute_a_decomposition</a>  )
 ;<br/>

<a NAME="E4:35"/><b>then </b><i><font color="Green">E12</font></i>: 
( <font color="Maroon">A1</font>,<font color="Maroon">B1</font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a>  &amp; <font color="Maroon">A2</font>,<font color="Maroon">B2</font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a>  )
 
<b>by </b><i><a class="ref" href="tsep_2.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E5:35"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E6:35"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="txt" href="tsep_2.html#E6:35"><i><font color="Green">E26</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E7:35"/>
( <font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <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> #) implies <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <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> #)
 ;<br/>

<a NAME="E2:35_1"/><b>then </b>
<font color="Maroon">B1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</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_1.html#D2">TSEP_1:def 2</a></i>;<br/>
<a NAME="E3:35_1"/><b>then </b>
<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>
 
;<br/>
<a NAME="E4:35_1"/><b>then </b>
<span class="p1">(<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#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T8">TOPS_1:8</a></i>;<br/>
<a NAME="E5:35_1"/><b>then </b>
<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_1.html#T53">XBOOLE_1:53</a></i>;<br/>
<a NAME="E6:35_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><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E7:35_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/>


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

<b>assume </b><a NAME="E8:35"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E9:35"/><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><a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>
<a NAME="E10:35"/><b>then </b>
<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="pre_topc.html#K1">{}</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="E11:35"/><b>then </b>
<span class="p1">(<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#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a></i>;<br/>
<a NAME="E12:35"/><b>then </b>
<font color="Maroon">B1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
;<br/>
<a NAME="E13:35"/><b>then </b>
<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>
 
;<br/>
<a NAME="E14:35"/><b>then </b>
<font color="Maroon">B1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E15:35"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</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="E16:35"/>
<font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <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> #)
 <b>by </b><i><a class="ref" href="tsep_1.html#T5">TSEP_1:5</a></i>;<br/>


</div>
