<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">E</font> =  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font>;<br/>
<b>consider </b><font color="Maroon">EqR</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">a</font><b> such that </b><br/><a NAME="E2:51_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">a</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">a</font> &amp;  ex <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">P</font>,<font color="Olive">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a>  ) ) )
 <b>by </b><i/>;<br/>
<a NAME="E3:51_1_1"/>
 <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">EqR</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<b>thus </b><a NAME="E1:51_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:51_1_1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> <b>by </b><i/>;<br/>
<a NAME="E4:51_1_1_1_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:51_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:51_1_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR</font>
 ;<br/>

<a NAME="E3:51_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> <b>by </b><i/>;<br/>
<a NAME="E5:51_1_1_1"/>
 ex <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> st <br/>( <font color="Olive">P</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">Q</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">y</font> &amp; <font color="Olive">P</font>,<font color="Olive">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:51_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:51_1_1"/>
( not  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font> is <a href="relat_2.html#V3">symmetric</a> &amp;  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font> is <a href="relat_2.html#V8">transitive</a> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#T16">EQREL_1:16</a>, <a class="ref" href="relat_1.html#T63">RELAT_1:63</a></i>;<br/>


</div>
