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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>assume </b><a NAME="E1:107"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 ;<br/>

<b>assume </b><a NAME="E2:107"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">X2</font>
 ;<br/>

<b>thus </b><a NAME="E3:107"/>
( <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a>  implies ( <font color="Maroon">X1</font> is  <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">X2</font> is  <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><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/>;<br/>
<a NAME="E3:107_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_1.html#D3" target="_self">Def3</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:107_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
;<br/>
<b>assume </b><a NAME="E5:107_1"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 ;<br/>

<a NAME="E6:107_1"/><b>then </b>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#D5" target="_self">Def5</a>, <a class="ref" href="tsep_1.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E7:107_1"/><b>then </b>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i><a class="txt" href="tsep_1.html#E21"><i><font color="Green">Lemma55</font></i></a></i>;<br/>
<a NAME="E8:107_1"/><b>then </b>
( <font color="Maroon">A1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A2</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T29" target="_self">Th29</a>, <a class="ref" href="connsp_1.html#T5">CONNSP_1:5</a></i>;<br/>
<b>hence </b><a NAME="E9:107_1"/>
( <font color="Maroon">X1</font> is  <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">X2</font> is  <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E4:107"/>
( <font color="Maroon">X1</font> is  <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">X2</font> is  <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> implies <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a>  )
 <b>by </b><i/>;<br/>


</div>
