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

<b>set </b><font color="Maroon">F</font> =  <a href="topalg_4.html#K12" target="_self">FGPrIso</a> <font color="Maroon">s</font>,<font color="Maroon">t</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>;<br/><b>consider </b><font color="Maroon">la</font> being   <a href="topalg_1.html#NM1">Loop</a> of <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span><b> such that </b><br/><a NAME="E2:43_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>,<font color="Maroon">la</font>
 <b>and </b><br/><a NAME="E3:43_1_1_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_4.html#K12" target="_self">FGPrIso</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p3">(<span class="default"><a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">la</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span>,<span class="p3">(<span class="default"><a href="topalg_4.html#K11" target="_self">pr2</a> <font color="Maroon">la</font></span>)</span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">lb</font> being   <a href="topalg_1.html#NM1">Loop</a> of <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span><b> such that </b><br/><a NAME="E5:43_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>,<font color="Maroon">lb</font>
 <b>and </b><br/><a NAME="E6:43_1_1_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_4.html#K12" target="_self">FGPrIso</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p3">(<span class="default"><a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">lb</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span>,<span class="p3">(<span class="default"><a href="topalg_4.html#K11" target="_self">pr2</a> <font color="Maroon">lb</font></span>)</span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">lab</font> being   <a href="topalg_1.html#NM1">Loop</a> of <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span><b> such that </b><br/><a NAME="E8:43_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>,<font color="Maroon">lab</font>
 <b>and </b><br/><a NAME="E9:43_1_1_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_4.html#K12" target="_self">FGPrIso</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p3">(<span class="default"><a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">lab</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span>,<span class="p3">(<span class="default"><a href="topalg_4.html#K11" target="_self">pr2</a> <font color="Maroon">lab</font></span>)</span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">A1</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">la</font></span>)</span>, <font color="Maroon">A2</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">lb</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span> <b>by </b><i><a class="ref" href="topalg_1.html#T48">TOPALG_1:48</a></i>;<br/><b>reconsider </b><font color="Maroon">B1</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_4.html#K11" target="_self">pr2</a> <font color="Maroon">la</font></span>)</span>, <font color="Maroon">B2</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_4.html#K11" target="_self">pr2</a> <font color="Maroon">lb</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span> <b>by </b><i><a class="ref" href="topalg_1.html#T48">TOPALG_1:48</a></i>;<br/><a NAME="E12:43_1_1_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">la</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">lb</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="topalg_4.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="topalg_1.html#T62">TOPALG_1:62</a></i>;<br/><a NAME="E13:43_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">lab</font>,<font color="Maroon">la</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">lb</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i>, <a class="ref" href="topalg_1.html#T47">TOPALG_1:47</a></i>;<br/><a NAME="E14:43_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">lab</font>, <a href="topalg_4.html#K10" target="_self">pr1</a> <span class="p1">(<span class="default"><font color="Maroon">la</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">lb</font></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i/>;<br/><a NAME="E15:43_1_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="topalg_4.html#K11" target="_self">pr2</a> <font color="Maroon">lab</font>, <a href="topalg_4.html#K11" target="_self">pr2</a> <span class="p1">(<span class="default"><font color="Maroon">la</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">lb</font></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i>, </i>;<br/><i><font color="Green">E76</font></i>: <font color="Maroon">A1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">A2</font> = 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">la</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">lb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topalg_1.html#T62">TOPALG_1:62</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_4.html#K10" target="_self">pr1</a> <span class="p2">(<span class="default"><font color="Maroon">la</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">lb</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">lab</font></span>)</span>
<b>by </b><i>, <a class="ref" href="topalg_1.html#T47">TOPALG_1:47</a></i>
;<br/><font color="Maroon">B1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">B2</font> = 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_4.html#K11" target="_self">pr2</a> <font color="Maroon">la</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="topalg_4.html#K11" target="_self">pr2</a> <font color="Maroon">lb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topalg_1.html#T62">TOPALG_1:62</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_4.html#K11" target="_self">pr2</a> <span class="p2">(<span class="default"><font color="Maroon">la</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">lb</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_4.html#K11" target="_self">pr2</a> <font color="Maroon">lab</font></span>)</span>
<b>by </b><i>, <a class="ref" href="topalg_1.html#T47">TOPALG_1:47</a></i>
;<br/><b>hence </b><a NAME="E18:43_1_1_1"/>
<span class="p1">(<span class="default"><a href="topalg_4.html#K12" target="_self">FGPrIso</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_4.html#K12" target="_self">FGPrIso</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_4.html#K12" target="_self">FGPrIso</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_4.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="topalg_4.html#T1" target="_self">Th1</a>, , <a class="txt" href="topalg_4.html#E2:43_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="group_7.html#T32">GROUP_7:32</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_1_1"/>
 <a href="topalg_4.html#K12" target="_self">FGPrIso</a> <font color="Maroon">s</font>,<font color="Maroon">t</font> is   <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K15">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p3">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span></span><a href="group_7.html#K15">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>


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