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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>;<br/>



<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="E3:105"/><i><font color="Green">E17</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</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="E4:105"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">X</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E5:105"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font>
 ;<br/>

<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_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="E2:105_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T16" target="_self">Th16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_1"/>
( <font color="Maroon">X1</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> )
 ;<br/><a NAME="E2:105_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( not <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; not <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">C1</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C2</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:105_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">C1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>and </b><br/><a NAME="E5:105_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 <b>and </b><br/><a NAME="E6:105_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> or  ex <font color="Olive">C</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> ) )
 <b>by </b><i><a class="ref" href="tsep_1.html#T30" target="_self">Th30</a>, <a class="ref" href="tsep_1.html#T10" target="_self">Th10</a></i>;<br/><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_1_1"/>
<font color="Maroon">C1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 ;<br/><a NAME="E2:105_1_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_1_1_1_1"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> or  not <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:105_1_1_1_1_1_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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#T28">XBOOLE_1:28</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T43" target="_self">Th43</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>

;<br/><a NAME="E3:105_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T40" target="_self">Th40</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:105_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tsep_1.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:105_1_1_1_1_1_2"/>
not <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">C</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:105_1_1_1_1_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 <b>and </b><br/><a NAME="E4:105_1_1_1_1_1_2"/><i><font color="Green">E96</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T41" target="_self">Th41</a></i>;<br/><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T43" target="_self">Th43</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/><a NAME="E6:105_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> <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> <font color="Maroon">A2</font></span>)</span> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T40" target="_self">Th40</a>, <a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:105_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>hence </b><a NAME="E8:105_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tsep_1.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:105_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><i><font color="Green">E118</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_1_2"/>
<font color="Maroon">C2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 ;<br/><a NAME="E2:105_1_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_1_2_1_1"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> or  not <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:105_1_1_2_1_1_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>

;<br/><a NAME="E3:105_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T40" target="_self">Th40</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:105_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tsep_1.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:105_1_1_2_1_1_2"/>
not <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">C</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:105_1_1_2_1_1_2"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 <b>and </b><br/><a NAME="E4:105_1_1_2_1_1_2"/><i><font color="Green">E121</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T41" target="_self">Th41</a></i>;<br/><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/><a NAME="E6:105_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <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> <font color="Maroon">A2</font></span>)</span> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T40" target="_self">Th40</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:105_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>hence </b><a NAME="E8:105_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tsep_1.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:105_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><b>thus </b><a NAME="E9:105_1_1"/>
 ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; ( <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>(  <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> <span class="p1">(<span class="default"><font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Y1</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:105_1_1_3"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, </i>;<br/>
<b>consider </b><font color="Maroon">Y2</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:105_1_1_3"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">Y1</font>
;<br/>

<b>take </b>
<font color="Maroon">Y2</font>
;<br/>

<a NAME="E5:105_1_1_3"/><i><font color="Green">E124</font></i>: 
( 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" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> &amp; 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" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E6:105_1_1_3"/><b>then </b>
( <font color="Maroon">Y1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E7:105_1_1_3"/><b>then </b><i><font color="Green">E125</font></i>: 
( 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#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, <a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_1_3_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font>
 ;<br/><a NAME="E2:105_1_1_3_1"/><b>then </b><i><font color="Green">E127</font></i>: 
not <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, </i>;<br/><b>consider </b><font color="Maroon">C</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:105_1_1_3_1"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T41" target="_self">Th41</a>, <a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, , </i>;<br/><b>thus </b><a NAME="E5:105_1_1_3_1"/>
 ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>(  <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> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Y</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:105_1_1_3_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T48" target="_self">Th48</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<i><font color="Green">E130</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> = 
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" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, <a class="ref" href="tsep_1.html#T48" target="_self">Th48</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T49" target="_self">Th49</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_3_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_1_3_1_1_2"/>
<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 ;<br/><a NAME="E2:105_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <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/><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T48" target="_self">Th48</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="tsep_1.html#T51" target="_self">Th51</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>

;<br/><b>hence </b><a NAME="E4:105_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tsep_1.html#T47" target="_self">Th47</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:105_1_1_3_1_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, <a class="ref" href="tsep_1.html#T49" target="_self">Th49</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:105_1_1_3_1_1"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> &amp; 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#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T29" target="_self">Th29</a>, <a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, <a class="ref" href="tsep_1.html#T49" target="_self">Th49</a>, </i>;<br/>
<b>hence </b><a NAME="E7:105_1_1_3_1_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> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#D5" target="_self">Def5</a>, <a class="ref" href="tsep_1.html#T48" target="_self">Th48</a>, <a class="ref" href="tsep_1.html#D7" target="_self">Def7</a>, , </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:105_1_1_3"/>
( <font color="Maroon">Y1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; ( <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>(  <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> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) ) )
 <b>by </b><i><a class="ref" href="tsep_1.html#T40" target="_self">Th40</a>, , </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:105_1"/>
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> or <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> or  ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; ( <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>(  <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> <span class="p1">(<span class="default"><font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) ) ) )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_2"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> or <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> or  ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; ( <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>(  <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> <span class="p1">(<span class="default"><font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) ) ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_2_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">X1</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> )
 ;<br/><b>then consider </b><font color="Maroon">Y1</font> being  non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">Y2</font> being  non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:105_2_1"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">Y1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> )
 <b>and </b><br/><a NAME="E4:105_2_1"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>(  <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> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) )
 <b>by </b><i><a class="ref" href="tsep_1.html#T30" target="_self">Th30</a></i>;<br/><b>reconsider </b><font color="Maroon">C1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y1</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">C2</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/>;<br/><a NAME="E7:105_2_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">C1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i/>;<br/><a NAME="E8:105_2_1"/><i><font color="Green">E137</font></i>: 
( 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" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> &amp; 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" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/><div><i><font color="Green">E138</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_2_1_1"/>
<font color="Maroon">Y1</font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 ;<br/><a NAME="E2:105_2_1_1"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">C1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_2_1_1_1_1"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font> or  not <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:105_2_1_1_1_1_1"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font>
 ;<br/><div class="add"><a NAME="E2:105_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><b>then </b><i><font color="Green">E140</font></i>: <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
<font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>

;<br/><a NAME="E4:105_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">C2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:105_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E6:105_2_1_1_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><b>hence </b><a NAME="E7:105_2_1_1_1_1_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:105_2_1_1_1_1_2"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Y</font> being  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:105_2_1_1_1_1_2"/><i><font color="Green">E141</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> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:105_2_1_1_1_1_2"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T40" target="_self">Th40</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E6:105_2_1_1_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/><b>then </b><i><font color="Green">E143</font></i>: <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/><div><i><font color="Green">E144</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_2_1_1_1_1_2_2"/>
<font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:105_2_1_1_1_1_2_2"/><b>then </b>
<font color="Maroon">C2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</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/><a NAME="E3:105_2_1_1_1_1_2_2"/><b>then </b>
<font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:105_2_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E145</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E5:105_2_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_1_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_2_1_1_1_1_2_2_1_1"/>
( <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:105_2_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:105_2_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, , <a class="ref" href="tsep_1.html#T47" target="_self">Th47</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:105_2_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:105_2_1_1_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</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/><a NAME="E3:105_2_1_1_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:105_2_1_1_1_1_2_2_1_1_2"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> &amp; 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#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T29" target="_self">Th29</a>, <a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E5:105_2_1_1_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, </i>;<br/><a NAME="E6:105_2_1_1_1_1_2_2_1_1_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> <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> <font color="Maroon">A2</font></span>)</span> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, <a class="ref" href="tsep_1.html#T48" target="_self">Th48</a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E7:105_2_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:105_2_1_1_1_1_2_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_1_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_2_1_1_1_1_2_3"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:105_2_1_1_1_1_2_3"/><b>then </b>
<font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</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/><a NAME="E3:105_2_1_1_1_1_2_3"/><b>then </b>
<font color="Maroon">Y</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:105_2_1_1_1_1_2_3"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> &amp; 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#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T29" target="_self">Th29</a>, <a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E5:105_2_1_1_1_1_2_3"/><b>then </b><i><font color="Green">E147</font></i>: 
( <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, , <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E6:105_2_1_1_1_1_2_3"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_1_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_1_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_2_1_1_1_1_2_3_1_1"/>
( <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_1_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:105_2_1_1_1_1_2_3_1_1_1"/>
<font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:105_2_1_1_1_1_2_3_1_1_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#T47" target="_self">Th47</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_1_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:105_2_1_1_1_1_2_3_1_1_2"/>
<font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:105_2_1_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">C2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</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/><a NAME="E3:105_2_1_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:105_2_1_1_1_1_2_3_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E5:105_2_1_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, </i>;<br/><b>hence </b><a NAME="E6:105_2_1_1_1_1_2_3_1_1_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#T48" target="_self">Th48</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:105_2_1_1_1_1_2_3"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:105_2_1_1_1_1_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:105_2_1_1"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:105_2_1_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:105_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tsep_1.html#T38" target="_self">Th38</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_2_1_2"/>
not <font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 ;<br/><a NAME="E2:105_2_1_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">C2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_2_1_2_1_1"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font> or  not <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:105_2_1_2_1_1_1"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font>
 ;<br/><div class="add"><a NAME="E2:105_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><b>then </b><i><font color="Green">E150</font></i>: <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <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></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">C1</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>

;<br/><a NAME="E4:105_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">C1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:105_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">Y1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E6:105_2_1_2_1_1_1"/><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#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><b>hence </b><a NAME="E7:105_2_1_2_1_1_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:105_2_1_2_1_1_2"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Y</font> being  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:105_2_1_2_1_1_2"/><i><font color="Green">E151</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> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:105_2_1_2_1_1_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T40" target="_self">Th40</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E6:105_2_1_2_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/><b>then </b><i><font color="Green">E153</font></i>: <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <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></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/><div><i><font color="Green">E154</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_2_1_2_1_1_2_2"/>
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:105_2_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">C1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</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/><a NAME="E3:105_2_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">Y1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:105_2_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E155</font></i>: 
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#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E5:105_2_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_2_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_2_1_2_1_1_2_2_1_1"/>
( <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:105_2_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:105_2_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, , <a class="ref" href="tsep_1.html#T47" target="_self">Th47</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:105_2_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:105_2_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</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/><a NAME="E3:105_2_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:105_2_1_2_1_1_2_2_1_1_2"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> &amp; 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#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T29" target="_self">Th29</a>, <a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E5:105_2_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, </i>;<br/><a NAME="E6:105_2_1_2_1_1_2_2_1_1_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <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> <font color="Maroon">A2</font></span>)</span> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, <a class="ref" href="tsep_1.html#T48" target="_self">Th48</a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E7:105_2_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:105_2_1_2_1_1_2_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_2_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_2_1_2_1_1_2_3"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:105_2_1_2_1_1_2_3"/><b>then </b>
<font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</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/><a NAME="E3:105_2_1_2_1_1_2_3"/><b>then </b>
<font color="Maroon">Y</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:105_2_1_2_1_1_2_3"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> &amp; 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#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T29" target="_self">Th29</a>, <a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E5:105_2_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E157</font></i>: 
( <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, , <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E6:105_2_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_2_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_2_1_2_1_1_2_3_1_1"/>
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:105_2_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:105_2_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#T47" target="_self">Th47</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:105_2_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:105_2_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">C1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</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/><a NAME="E3:105_2_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">Y1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:105_2_1_2_1_1_2_3_1_1_2"/><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#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E5:105_2_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, </i>;<br/><b>hence </b><a NAME="E6:105_2_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#T48" target="_self">Th48</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:105_2_1_2_1_1_2_3"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:105_2_1_2_1_1_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:105_2_1_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:105_2_1_2"/><b>then </b>
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:105_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tsep_1.html#T38" target="_self">Th38</a>, </i>;<br/></div><b>end;</b></div><a NAME="E11:105_2_1"/><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#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T43" target="_self">Th43</a>, </i>;<br/><a NAME="E12:105_2_1"/><b>then </b><i><font color="Green">E159</font></i>: 
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T39" target="_self">Th39</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_2_1_3_1"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> or  not <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_3_1_1"><b>suppose </b></a><a NAME="E1:105_2_1_3_1_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:105_2_1_3_1_1"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/>

<a NAME="E1:105_2_1_3_1_1_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T41" target="_self">Th41</a>, <a class="ref" href="tops_1.html#T36">TOPS_1:36</a></i>;<br/>
<a NAME="E2:105_2_1_3_1_1_1"/>
<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>
<a NAME="E3:105_2_1_3_1_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <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="E4:105_2_1_3_1_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#D3" target="_self">Def3</a>, <a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, <a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_3_1_2"><b>suppose </b></a><a NAME="E1:105_2_1_3_1_2"/><i><font color="Green">E162</font></i>: 
not <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:105_2_1_3_1_2"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_3_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Y</font> being  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:105_2_1_3_1_2_1"/><i><font color="Green">E163</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> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T40" target="_self">Th40</a>, <a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E4:105_2_1_3_1_2_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="tsep_1.html#E3:105"><i><font color="Green">E17</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1_3_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_2_1_3_1_2_1_1"/>
not <font color="Maroon">Y</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font>
 ;<br/><a NAME="E2:105_2_1_3_1_2_1_1"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E3:105_2_1_3_1_2_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, <a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/><b>then </b><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <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#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <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></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"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <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></span>)</span> <a href="xboole_0.html#K2">\/</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">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>

;<br/><b>hence </b><a NAME="E5:105_2_1_3_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tsep_1.html#T45" target="_self">Th45</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:105_2_1_3_1_2_1"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> &amp; 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#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T29" target="_self">Th29</a>, <a class="ref" href="tsep_1.html#T42" target="_self">Th42</a>, </i>;<br/>
<a NAME="E7:105_2_1_3_1_2_1"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, </i>;<br/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> = 
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" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self">Th46</a>, <a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>
;<br/>
<b>hence </b><a NAME="E9:105_2_1_3_1_2_1"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E14:105_2_1"/><b>then </b>
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_1.html#R2" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T41" target="_self">Th41</a>, <a class="ref" href="tsep_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E15:105_2_1"/>
<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><a class="ref" href="tsep_1.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:105_2"/>
<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><a class="ref" href="tsep_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:105"/>
( <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a>  iff ( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> or <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> or  ex <font color="Olive">Y1</font>, <font color="Olive">Y2</font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">Y2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; ( <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font> or  ex <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>(  <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> <span class="p1">(<span class="default"><font color="Olive">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y2</font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> ) ) ) ) )
 <b>by </b><i/>;<br/>


</div>
