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

<b>let </b><font color="Maroon" title="c3">E</font>, <font color="Maroon" title="c4">F</font> be   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of  <a href="topalg_1.html#K1" title="TOPALG_1:func.1">Loops</a> <font color="Maroon" title="c2">a</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_1_2"/><i><font color="Green" title="E31">A2</font></i>: 
for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> iff <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )
 <b>and </b><br/><a NAME="E2:50_1_2"/><i><font color="Green" title="E32">A3</font></i>: 
for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> iff <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</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="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a><br/>

<b>thus </b><a NAME="E3:50_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_2_1"/><i><font color="Green" title="E33">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font>
 ;<br/>

<a NAME="E2:50_1_2_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topalg_1.html#K1" title="TOPALG_1:func.1">Loops</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topalg_1.html#K1" title="TOPALG_1:func.1">Loops</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">y</font> as   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font> <b>by </b><i><a class="ref" href="topalg_1.html#D1" target="_self" title="TOPALG_1:def.1">Def1</a></i>;<br/>
<a NAME="E4:50_1_2_1"/>
<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:50_1_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="topalg_1.html#E1:50_1_2_1"><i><font color="Green" title="E33">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:50_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:50_1_2"><i><font color="Green" title="E32">A3</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E4:50_1_2"/><i><font color="Green" title="E33">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 ;<br/>

<a NAME="E5:50_1_2"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topalg_1.html#K1" title="TOPALG_1:func.1">Loops</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topalg_1.html#K1" title="TOPALG_1:func.1">Loops</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">y</font> as   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font> <b>by </b><i><a class="ref" href="topalg_1.html#D1" target="_self" title="TOPALG_1:def.1">Def1</a></i>;<br/>
<a NAME="E7:50_1_2"/>
<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:50_1_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="topalg_1.html#E4:50_1_2"><i><font color="Green" title="E33">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:50_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:50_1_2"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>


</div>
