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<b>consider </b><font color="Maroon">E</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="E2:50_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">E</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/>
<b>take </b>
<font color="Maroon">E</font>
;<br/>

<b>let </b><font color="Maroon">P</font> be   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font>, <font color="Maroon">Q</font> be   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font>;<br/>

<b>thus </b><a NAME="E3:50_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">Q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> implies <font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">Q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/>

<a NAME="E2:50_1_1_1"/><b>then </b>
 ex <font color="Olive">P1</font>, <font color="Olive">Q1</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> st <br/>( <font color="Olive">P1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">P</font> &amp; <font color="Olive">Q1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">Q</font> &amp; <font color="Olive">P1</font>,<font color="Olive">Q1</font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:50_1_1_1"/>
<font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 ;<br/>


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

<a NAME="E4:50_1_1"/>
( <font color="Maroon">P</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">Q</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/>;<br/>
<b>hence </b><a NAME="E5:50_1_1"/>
( <font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a>  implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">Q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 <b>by </b><i/>;<br/>


</div>
