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

<b>set </b><font color="Maroon" title="c8">pS</font> =  <a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c6">s</font>;<br/>
<b>set </b><font color="Maroon" title="c9">h</font> =  <a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</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="c4">S</font>,<font color="Maroon" title="c6">s</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font></span>)</span></span><br/><b>consider </b><font color="Maroon" title="c12">lsa</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c6">s</font><b> such that </b><br/><a NAME="E2:39_1_1_1"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="c4">S</font>,<font color="Maroon" title="c6">s</font></span>)</span>,<font color="Maroon" title="c12">lsa</font>
 <b>and </b><br/><a NAME="E3:39_1_1_1"/><i><font color="Green" title="E16">A2</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="c5">T</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <font color="Maroon" title="c12">lsa</font></span>)</span>
 <b>by </b><i><a class="ref" href="topalg_3.html#D1" target="_self" title="TOPALG_3:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">lsb</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c6">s</font><b> such that </b><br/><a NAME="E5:39_1_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="c4">S</font>,<font color="Maroon" title="c6">s</font></span>)</span>,<font color="Maroon" title="c13">lsb</font>
 <b>and </b><br/><a NAME="E6:39_1_1_1"/><i><font color="Green" title="E18">A4</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="c5">T</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <font color="Maroon" title="c13">lsb</font></span>)</span>
 <b>by </b><i><a class="ref" href="topalg_3.html#D1" target="_self" title="TOPALG_3:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">lsab</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c6">s</font><b> such that </b><br/><a NAME="E8:39_1_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="c4">S</font>,<font color="Maroon" title="c6">s</font></span>)</span>,<font color="Maroon" title="c14">lsab</font>
 <b>and </b><br/><a NAME="E9:39_1_1_1"/><i><font color="Green" title="E20">A6</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="c5">T</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <font color="Maroon" title="c14">lsab</font></span>)</span>
 <b>by </b><i><a class="ref" href="topalg_3.html#D1" target="_self" title="TOPALG_3:def.1">Def1</a></i>;<br/><a NAME="E10:39_1_1_1"/><i><font color="Green" title="E21">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <font color="Maroon" title="c12">lsa</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <font color="Maroon" title="c13">lsb</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">lsa</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">lsb</font></span>)</span>
 <b>by </b><i><a class="ref" href="topalg_3.html#T30" target="_self" title="TOPALG_3:th.30">Th30</a></i>;<br/><a NAME="E11:39_1_1_1"/>
<font color="Maroon" title="c10">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="c4">S</font>,<font color="Maroon" title="c6">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">lsa</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">lsb</font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:39_1_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="topalg_3.html#E5:39_1_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="topalg_1.html#T62" title="TOPALG_1:th.62">TOPALG_1:62</a></i>;<br/><a NAME="E12:39_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">lsab</font>,<font color="Maroon" title="c12">lsa</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">lsb</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_3.html#E8:39_1_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="topalg_1.html#T47" title="TOPALG_1:th.47">TOPALG_1:47</a></i>;<br/><a NAME="E13:39_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <font color="Maroon" title="c14">lsab</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <font color="Maroon" title="c12">lsa</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <font color="Maroon" title="c13">lsb</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_3.html#E10:39_1_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="topalg_3.html#T28" target="_self" title="TOPALG_3:th.28">Th28</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">b</font></span>)</span> = 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="c5">T</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <font color="Maroon" title="c12">lsa</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="topalg_3.html#K2" title="TOPALG_3:func.2">*</a> <font color="Maroon" title="c13">lsb</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E9:39_1_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="topalg_1.html#T47" title="TOPALG_1:th.47">TOPALG_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E3:39_1_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="topalg_3.html#E6:39_1_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="topalg_1.html#T62" title="TOPALG_1:th.62">TOPALG_1:62</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1"/>
 <a href="topalg_3.html#K3" title="TOPALG_3:func.3">FundGrIso</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">s</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</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="c4">S</font>,<font color="Maroon" title="c6">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c5">T</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
