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

<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font>;<br/>
<b>set </b><font color="Maroon" title="c4">g</font> = the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">E</font> =  <a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font>;<br/>
<b>thus </b><a NAME="E1:59_1_1"/>
 <a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a><br/>

<b>consider </b><font color="Maroon" title="c9">A</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E2:59_1_1_1"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c9">A</font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self" title="TOPALG_1:th.48">Th48</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">B</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E4:59_1_1_1"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self" title="TOPALG_1:th.48">Th48</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">C</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E6:59_1_1_1"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c11">C</font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self" title="TOPALG_1:th.48">Th48</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">AB</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E8:59_1_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c12">AB</font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self" title="TOPALG_1:th.48">Th48</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">BC</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E10:59_1_1_1"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c13">BC</font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self" title="TOPALG_1:th.48">Th48</a></i>;<br/>
<a NAME="E11:59_1_1_1"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T15" title="BORSUK_2:th.15">BORSUK_2:15</a></i>;<br/>
<a NAME="E12:59_1_1_1"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c11">C</font>,<font color="Maroon" title="c11">C</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T15" title="BORSUK_2:th.15">BORSUK_2:15</a></i>;<br/>
<a NAME="E13:59_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:59_1_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="topalg_1.html#E4:59_1_1_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="topalg_1.html#E54"><i><font color="Green" title="E36">Lm3</font></i></a></i>;<br/>
<a NAME="E14:59_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">AB</font>,<font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">B</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#E8:59_1_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>;<br/>
<a NAME="E15:59_1_1_1"/><b>then </b><i><font color="Green" title="E44">A8</font></i>: 
<font color="Maroon" title="c12">AB</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">C</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#E12:59_1_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="borsuk_6.html#T83" title="BORSUK_6:th.83">BORSUK_6:83</a></i>;<br/>
<a NAME="E16:59_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E4:59_1_1_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="topalg_1.html#E6:59_1_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="topalg_1.html#E54"><i><font color="Green" title="E36">Lm3</font></i></a></i>;<br/>
<a NAME="E17:59_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">BC</font>,<font color="Maroon" title="c10">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">C</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#E10:59_1_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>;<br/>
<a NAME="E18:59_1_1_1"/><b>then </b><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">BC</font>,<font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">C</font></span>)</span> <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#E11:59_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="borsuk_6.html#T83" title="BORSUK_6:th.83">BORSUK_6:83</a></i>;<br/>
<a NAME="E19:59_1_1_1"/><i><font color="Green" title="E46">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">C</font>,<font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T81" title="BORSUK_6:th.81">BORSUK_6:81</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">z</font> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">AB</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">C</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E6:59_1_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="topalg_1.html#E8:59_1_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="topalg_1.html#E54"><i><font color="Green" title="E36">Lm3</font></i></a></i>
<br/>.= 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">C</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E15:59_1_1_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E19:59_1_1_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">BC</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E18:59_1_1_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>
<br/>.= 
<font color="Maroon" title="c6">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">z</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E2:59_1_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="topalg_1.html#E10:59_1_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="topalg_1.html#E54"><i><font color="Green" title="E36">Lm3</font></i></a></i>
;<br/>


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

<b>consider </b><font color="Maroon" title="c6">C</font> being  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font>;<br/>
<b>set </b><font color="Maroon" title="c7">e</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c6">C</font>;<br/>
<a NAME="E3:59_1_1"/>
<font color="Maroon" title="c6">C</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="topalg_1.html#D1" target="_self" title="TOPALG_1:def.1">Def1</a></i>;<br/>
<a NAME="E4:59_1_1"/><b>then </b>
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">e</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c6">C</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span> <b>by </b><i><a class="ref" href="topalg_1.html#D3" target="_self" title="TOPALG_1:def.3">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon" title="c9">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c10">A</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E7:59_1_1"/><i><font color="Green" title="E37">A11</font></i>: 
<font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c10">A</font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self" title="TOPALG_1:th.48">Th48</a></i>;<br/>
<a NAME="E8:59_1_1"/><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c10">A</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T88" title="BORSUK_6:th.88">BORSUK_6:88</a></i>;<br/>
<a NAME="E9:59_1_1"/><i><font color="Green" title="E39">A13</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">A</font>,<font color="Maroon" title="c10">A</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T90" title="BORSUK_6:th.90">BORSUK_6:90</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c9">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">e</font> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c6">C</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E7:59_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="topalg_1.html#E54"><i><font color="Green" title="E36">Lm3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">h</font>
<b>by </b><i><a class="txt" href="topalg_1.html#E7:59_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="topalg_1.html#E8:59_1_1"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c8">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">h</font> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">A</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E7:59_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="topalg_1.html#E54"><i><font color="Green" title="E36">Lm3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">h</font>
<b>by </b><i><a class="txt" href="topalg_1.html#E7:59_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="topalg_1.html#E9:59_1_1"><i><font color="Green" title="E39">A13</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>
;<br/>

<b>set </b><font color="Maroon" title="c11">x</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c10">A</font></span>)</span>;<br/>
<a NAME="E12:59_1_1"/>
 <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c10">A</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="topalg_1.html#D1" target="_self" title="TOPALG_1:def.1">Def1</a></i>;<br/>
<a NAME="E13:59_1_1"/><b>then </b>
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">x</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c10">A</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span> <b>by </b><i><a class="ref" href="topalg_1.html#D3" target="_self" title="TOPALG_1:def.3">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">x</font>
;<br/>

<a NAME="E15:59_1_1"/><i><font color="Green" title="E40">A14</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c10">A</font></span>)</span>,<font color="Maroon" title="c6">C</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T92" title="BORSUK_6:th.92">BORSUK_6:92</a></i>;<br/>
<a NAME="E16:59_1_1"/><i><font color="Green" title="E41">A15</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">A</font>,<font color="Maroon" title="c6">C</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T94" title="BORSUK_6:th.94">BORSUK_6:94</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c9">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">x</font> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c10">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E7:59_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="topalg_1.html#E54"><i><font color="Green" title="E36">Lm3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">e</font>
<b>by </b><i><a class="txt" href="topalg_1.html#E15:59_1_1"><i><font color="Green" title="E40">A14</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c12">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">h</font> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">A</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E7:59_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="topalg_1.html#E54"><i><font color="Green" title="E36">Lm3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">e</font>
<b>by </b><i><a class="txt" href="topalg_1.html#E16:59_1_1"><i><font color="Green" title="E41">A15</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>
;<br/>


</div>
