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<b>let </b><font color="Maroon">GX</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">GY</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">GX</font>,<font color="Maroon">GY</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GY</font>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">GX</font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 ;<br/>

<a NAME="E4:16"/><i><font color="Green">E27</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GY</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GX</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E5:16"/>
not <font color="Maroon">GY</font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GY</font>, <font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GY</font><b> such that </b><br/><a NAME="E7:16"/><i><font color="Green">E29</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GY</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:16"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GY</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GY</font> )
 <b>and </b><br/><a NAME="E9:16"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>and </b><br/><a NAME="E10:16"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:16"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="connsp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E12:16"/><i><font color="Green">E46</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GY</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<i><font color="Green">E47</font></i>:  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> = 
<font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GY</font>
<b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T48">FUNCT_2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T175">RELAT_1:175</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font></span>)</span> = 
<font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="connsp_1.html#E1:16"><i><font color="Green">E22</font></i></a>, <a class="ref" href="relat_1.html#T171">RELAT_1:171</a></i>
;<br/>
<a NAME="E15:16"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E16:16"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#D1" target="_self">Def1</a>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>
<a NAME="E17:16"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GX</font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T45">FUNCT_2:45</a></i>;<br/>
<a NAME="E18:16"/><b>then </b>
( <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font> &amp; <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font> )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T174">RELAT_1:174</a></i>;<br/>
<b>hence </b><a NAME="E19:16"/>
contradiction
 <b>by </b><i>, <a class="txt" href="connsp_1.html#E2:16"><i><font color="Green">E24</font></i></a>, <a class="txt" href="connsp_1.html#E3:16"><i><font color="Green">E25</font></i></a>, <a class="ref" href="connsp_1.html#T4" target="_self">Th4</a>, </i>;<br/>


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