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<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>assume </b><a NAME="E1:62"/><i><font color="Green">E21</font></i>: 
 <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span></span>)</span>;<br/>
<b>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> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:62"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:62"/><b>then </b><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="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E6:62"/><i><font color="Green">E36</font></i>: 
 <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:62"/><b>then </b><i><font color="Green">E37</font></i>: 
<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"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:62"/>
 <a href="tops_1.html#K2" target="_self">Fr</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="tarski.html#R1">c=</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E9:62"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:62"/>
<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#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="tops_1.html#T40" target="_self">Th40</a>, <a class="ref" href="tops_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E11:62"/><b>then </b><i><font color="Green">E39</font></i>: 
<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#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span> <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="E12:62"/><i><font color="Green">E40</font></i>: 
<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#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <span class="p3">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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#K3">`</a> </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">K</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E13:62"/><i><font color="Green">E41</font></i>: 
 <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">K</font> <a href="xboole_0.html#R1">misses</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#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E14:62"/>
<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#K3">`</a> </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">K</font></span>)</span> <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="tops_1.html#T43" target="_self">Th43</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E15:62"/>
contradiction
 <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/>


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