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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">W</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">B</font> being  <a href="subset_1.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>
 ;<br/>

<a NAME="E2:28"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E3:28"/><b>then </b>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K2" target="_self">proj</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:28"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <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#K2" 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="funct_1.html#T146">FUNCT_1:146</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">e</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="E5:28"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <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#K2" 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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E7:28"/>
<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="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <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#K10">.:</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:28"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <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="funct_2.html#K3">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <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="funct_2.html#K3">.</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <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="funct_2.html#K3">.</a> <font color="Maroon">c</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:28"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <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="funct_2.html#K3">.</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:28"/><b>then </b>
<font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E13:28"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">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#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E14:28"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T12" target="_self">Th12</a>, </i>;<br/>
<b>hence </b><a NAME="E15:28"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>


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