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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">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> 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="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">P</font>,<font color="Olive">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a>  );<br/>
<a NAME="E1:49"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_1"/>
<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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> <b>by </b><i/>;<br/>
<a NAME="E3:49_1"/>
<font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T15">BORSUK_2:15</a></i>;<br/>
<b>hence </b><a NAME="E4:49_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">x</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:49"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>]
 
;<br/>
<a NAME="E3:49"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><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> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 ;<br/>

<b>then consider </b><font color="Maroon">P</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font>, <font color="Maroon">Q</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font><b> such that </b><br/><a NAME="E3:49_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 ;<br/>
<b>assume </b><a NAME="E4:49_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 ;<br/>

<b>then consider </b><font color="Maroon">R</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font>, <font color="Maroon">S</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font><b> such that </b><br/><a NAME="E6:49_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">R</font>,<font color="Maroon">S</font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 ;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E7:49_2"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">P</font>,<font color="Maroon">S</font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 <b>by </b><i>, , <a class="ref" href="borsuk_6.html#T87">BORSUK_6:87</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:49"/>
 ex <font color="Olive">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> st <br/>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="Olive">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; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1">EQREL_1:sch 1</a>(, , );</i><br/>


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