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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">D</font> be    <a href="borsuk_1.html#M1" target="_self">u.s.c._decomposition</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">D</font></span>)</span>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M2">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span>;<br/>







<a NAME="E1:90"/><i><font color="Green">E24</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:90"/>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T23" target="_self">Th23</a>, <a class="ref" href="borsuk_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:90"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:90"/><i><font color="Green">E31</font></i>: 
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 <b>and </b><br/><a NAME="E6:90"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Olive">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> holds <br/><font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T23" target="_self">Th23</a>, <a class="ref" href="borsuk_1.html#T21" target="_self">Th21</a></i>;<br/>
<b>set </b><font color="Maroon">Q</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>;<br/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K2" target="_self">proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_1.html#T23" target="_self">Th23</a>, <a class="ref" href="borsuk_1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<font color="Maroon">W</font>
<b>by </b><i><a class="ref" href="borsuk_1.html#T26" target="_self">Th26</a>, <a class="txt" href="borsuk_1.html#E1:90"><i><font color="Green">E24</font></i></a></i>
;<br/>
<a NAME="E8:90"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T24" target="_self">Th24</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:90"/><b>then </b>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T23" target="_self">Th23</a>, </i>;<br/>
<a NAME="E10:90"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E11:90"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T25" target="_self">Th25</a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E12:90"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E13:90"/>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T25" target="_self">Th25</a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E14:90"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>
<a NAME="E15:90"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E16:90"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T27" target="_self">Th27</a>, , <a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>
<b>hence </b><a NAME="E17:90"/>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>


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