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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">D</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c4">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>







<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<a NAME="E2:20"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:20"/>
<font color="Maroon" title="c4">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="borsuk_1.html#T18" target="_self" title="BORSUK_1:th.18">Th18</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c5">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:20"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:20"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">u</font> &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:20"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E6:20"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E8:20"/>
not <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<a NAME="E9:20"/><b>then </b>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E6:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="borsuk_1.html#E7:20"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:20"/><i><font color="Green" title="E21">A4</font></i>: 
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> &amp; <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E12:20"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E11:20"><i><font color="Green" title="E21">A4</font></i></a></i>;<br/>
<a NAME="E13:20"/>
not <font color="Maroon" title="c6">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">v</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E6:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="borsuk_1.html#E11:20"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E14:20"/><b>then </b>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E7:20"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="borsuk_1.html#E12:20"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E15:20"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_1.html#E6:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="borsuk_1.html#E11:20"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


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