<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">TS</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>







<b>assume </b><a NAME="E1:54"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x''</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">TS</font> ;<br/>
<a NAME="E3:54"/><i><font color="Green">E24</font></i>: 
not <font color="Maroon">TS</font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="txt" href="tops_1.html#E6:54"><i><font color="Green">E37</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:54"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E5:54"/><b>then </b><i><font color="Green">E36</font></i>: 
( not <font color="Maroon">x''</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span> &amp; not <font color="Maroon">x''</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:54"/><b>then </b><i><font color="Green">E37</font></i>: 
( not <font color="Maroon">x''</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span> &amp; not <font color="Maroon">x''</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span> &amp; not <font color="Maroon">x''</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">Q1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E8:54"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Q1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E9:54"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x''</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 <b>and </b><br/><a NAME="E10:54"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q1</font> or <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q1</font> )
 <b>by </b><i><a class="txt" href="tops_1.html#E8:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tops_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E11:54"/><i><font color="Green">E41</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="tops_1.html#E15:54"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E12:54"/>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a></i>;<br/>
<a NAME="E13:54"/><b>then </b><i><font color="Green">E42</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="tops_1.html#E17:54"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E14:54"/>
<font color="Maroon">K</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q1</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E6:54"><i><font color="Green">E37</font></i></a>, <a class="txt" href="tops_1.html#E8:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="tops_1.html#E11:54"><i><font color="Green">E41</font></i></a>, <a class="txt" href="tops_1.html#E13:54"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tops_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E15:54"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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>consider </b><font color="Maroon">Q2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E17:54"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Q2</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E18:54"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x''</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font>
 <b>and </b><br/><a NAME="E19:54"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q2</font> or <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q2</font> )
 <b>by </b><i><a class="txt" href="tops_1.html#E8:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="tops_1.html#E10:54"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tops_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E20:54"/><i><font color="Green">E73</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" target="_self">Th44</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E21:54"/>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>;<br/>
<a NAME="E22:54"/><b>then </b><i><font color="Green">E74</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E23:54"/>
<font color="Maroon">K</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q2</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E6:54"><i><font color="Green">E37</font></i></a>, <a class="txt" href="tops_1.html#E8:54"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="tops_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tops_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E24:54"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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/>
<a NAME="E25:54"/>
not <font color="Maroon">x''</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E6:54"><i><font color="Green">E37</font></i></a>, <a class="txt" href="tops_1.html#E9:54"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">Q3</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E27:54"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">Q3</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E28:54"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x''</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q3</font>
 <b>and </b><br/><a NAME="E29:54"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">L</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q3</font> or <font color="Maroon">L</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q3</font> )
 <b>by </b><i><a class="txt" href="tops_1.html#E8:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tops_1.html#T34" target="_self">Th34</a></i>;<br/>
<div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon">L</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:54_1"/><b>then </b>
<font color="Maroon">Q3</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E3:54_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">Q3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><a NAME="E4:54_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T40" target="_self">Th40</a>, , <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/><a NAME="E5:54_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tops_1.html#T53" target="_self">Th53</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E6:54_1"/>
<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</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_1.html#T16">XBOOLE_1:16</a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:54"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="tops_1.html#E11:54"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tops_1.html#T47" target="_self">Th47</a>, </i>;<br/>
<a NAME="E32:54"/>
<font color="Maroon">x''</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E13:54"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tops_1.html#T48" target="_self">Th48</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E33:54"/><b>then </b>
<font color="Maroon">K</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E6:54"><i><font color="Green">E37</font></i></a>, <a class="txt" href="tops_1.html#E8:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tops_1.html#T53" target="_self">Th53</a>, <a class="ref" href="tops_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E34:54"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">Q3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T49" target="_self">Th49</a>, <a class="ref" href="tops_1.html#T52" target="_self">Th52</a>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E35:54"/>
<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T45" target="_self">Th45</a>, <a class="ref" href="tops_1.html#T46" target="_self">Th46</a>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E36:54"/><b>then </b>
<font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</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_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E37:54"/><b>then </b>
<font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E38:54"/><b>then </b>
<font color="Maroon">K</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</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_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E39:54"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">K</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E40:54"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T47" target="_self">Th47</a>, </i>;<br/>
<a NAME="E41:54"/>
<font color="Maroon">x''</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q3</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tops_1.html#T48" target="_self">Th48</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E42:54"/>
contradiction
 <b>by </b><i><a class="txt" href="tops_1.html#E6:54"><i><font color="Green">E37</font></i></a>, <a class="txt" href="tops_1.html#E8:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tops_1.html#T54" target="_self">Th54</a>, <a class="ref" href="tops_1.html#T55" target="_self">Th55</a>, <a class="ref" href="tops_1.html#T34" target="_self">Th34</a></i>;<br/>


</div>
