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

<b>set </b><font color="Maroon">pS</font> =  <a href="topalg_1.html#NK4" target="_self">pi_1</a> <span class="p1"><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="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>;<br/>
<b>set </b><font color="Maroon">G</font> = <span class="p1"><a href="group_7.html#K15">&lt;*</a><span class="default"><span class="p2">(<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="p2">(<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>;<br/>
<b>set </b><font color="Maroon">pT</font> =  <a href="group_7.html#K2">product</a> <span class="p1"><a href="group_7.html#K15">&lt;*</a><span class="default"><span class="p2">(<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="p2">(<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>;<br/>
<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/>
<b>thus </b><a NAME="E1:42_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="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <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>
 ;<br/>

<b>then 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="E3:42_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="E4:42_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_1.html#K1">.</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>assume </b><a NAME="E5:42_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <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>
 ;<br/>

<b>then 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="E7:42_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="E8:42_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_1.html#K1">.</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>assume </b><a NAME="E9:42_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_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <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_1.html#K1">.</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E10:42_1_1_1"/><b>then </b>
(  <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> <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> <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> &amp;  <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> <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> <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> )
 
<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="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<a NAME="E11:42_1_1_1"/><b>then </b>
(  <a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">la</font>, <a href="topalg_4.html#K10" target="_self">pr1</a> <font color="Maroon">lb</font> <a href="borsuk_2.html#R3">are_homotopic</a>  &amp;  <a href="topalg_4.html#K11" target="_self">pr2</a> <font color="Maroon">la</font>, <a href="topalg_4.html#K11" target="_self">pr2</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="E12:42_1_1_1"/><b>then </b>
<font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:42_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <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#T47">TOPALG_1:47</a></i>;<br/>


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

<b>thus </b><a NAME="E2:42_1_1"/>
 <a href="relset_1.html#K5">rng</a> <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="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a><br/>

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:42_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <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>
 ;<br/>

<a NAME="E4:42_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <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> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</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_7.html#D2">GROUP_7:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:42_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E7:42_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</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>and </b><br/><a NAME="E8:42_1_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</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> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</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> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E9:42_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="txt" href="topalg_4.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E11:42_1_1"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="txt" href="topalg_4.html#E2"><i><font color="Green">Lemma32</font></i></a>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:42_1_1"/><i><font color="Green">E77</font></i>: 
 ex <font color="Olive">R1</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K15">&lt;*</a><span class="default"><span class="p2">(<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="p2">(<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> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</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> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> )
 
<b>by </b><i>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E13:42_1_1"/><i><font color="Green">E99</font></i>: 
 ex <font color="Olive">R2</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K15">&lt;*</a><span class="default"><span class="p2">(<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="p2">(<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> <a href="funct_1.html#K1">.</a> 2 &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</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> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> )
 
<b>by </b><i>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E14:42_1_1"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="group_7.html#K15">&lt;*</a><span class="default"><span class="p2">(<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="p2">(<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> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">S</font>,<font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E15:42_1_1"/><i><font color="Green">E132</font></i>: 
<span class="p1"><a href="group_7.html#K15">&lt;*</a><span class="default"><span class="p2">(<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="p2">(<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> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E16:42_1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</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> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="ref" href="topalg_4.html#T1" target="_self">Th1</a>, , </i>;<br/>
<b>then consider </b><font color="Maroon">l1</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">s</font><b> such that </b><br/><a NAME="E18:42_1_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <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> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">l1</font>
 <b>by </b><i>, , <a class="ref" href="topalg_1.html#T48">TOPALG_1:48</a></i>;<br/>
<a NAME="E19:42_1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</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> <a href="funct_1.html#K1">.</a> 2
 
<b>by </b><i><a class="ref" href="topalg_4.html#T1" target="_self">Th1</a>, , </i>;<br/>
<b>then consider </b><font color="Maroon">l2</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">t</font><b> such that </b><br/><a NAME="E21:42_1_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 2 <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> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span>,<font color="Maroon">l2</font>
 <b>by </b><i>, , <a class="ref" href="topalg_1.html#T48">TOPALG_1:48</a></i>;<br/>
<b>set </b><font color="Maroon">x</font> =  <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"><a href="topalg_4.html#K7" target="_self">&lt;:</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="topalg_4.html#K7" target="_self">:&gt;</a></span>;<br/>
<a NAME="E22:42_1_1"/>
 <a href="relset_1.html#K4">dom</a> <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="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E23:42_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
 <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"><a href="topalg_4.html#K7" target="_self">&lt;:</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="topalg_4.html#K7" target="_self">:&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <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>
 
<b>by </b><i><a class="ref" href="topalg_1.html#T48">TOPALG_1:48</a></i>;<br/>
<i><font color="Green">E143</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">l1</font> = 
the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">l2</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<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_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p3"><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="p3"><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="p2"><a href="topalg_4.html#K7" target="_self">&lt;:</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="topalg_4.html#K7" target="_self">:&gt;</a></span></span>)</span> = 
<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> <span class="p4"><a href="topalg_4.html#K7" target="_self">&lt;:</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="topalg_4.html#K7" target="_self">:&gt;</a></span></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> <span class="p4"><a href="topalg_4.html#K7" target="_self">&lt;:</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="topalg_4.html#K7" target="_self">:&gt;</a></span></span>)</span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
<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>,<font color="Maroon">l1</font></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> <span class="p4"><a href="topalg_4.html#K7" target="_self">&lt;:</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="topalg_4.html#K7" target="_self">:&gt;</a></span></span>)</span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
<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>,<font color="Maroon">l1</font></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>,<font color="Maroon">l2</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="topalg_4.html#E1"><i><font color="Green">Lemma31</font></i></a>, , , , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<b>hence </b><a NAME="E26:42_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <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>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div>
