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<span class="kw">set </span><font color="Maroon" title="c8">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/>
<span class="kw">set </span><font color="Maroon" title="c9">pS</font> =  <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c6">s</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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#NK6" title="TOPALG_1:NK.6">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:41_1_1_1"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<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#K4" title="TOPALG_1:func.4">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>
 <span class="kw">and </span><br/><a NAME="E3:41_1_1_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<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#K4" title="TOPALG_1:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_3.html#D1" target="_self" title="TOPALG_3:def.1">Def1</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:41_1_1_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<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#K4" title="TOPALG_1:func.4">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>
 <span class="kw">and </span><br/><a NAME="E6:41_1_1_1"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<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#K4" title="TOPALG_1:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_3.html#D1" target="_self" title="TOPALG_3:def.1">Def1</a></span>;<br/><a NAME="E7:41_1_1_1"/><span class="lab"><font color="Green" title="E19">A5</font></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> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_3.html#T30" target="_self" title="TOPALG_3:th.30">Th30</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E9:41_1_1_1"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<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#K4" title="TOPALG_1:func.4">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>
 <span class="kw">and </span><br/><a NAME="E10:41_1_1_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<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#K4" title="TOPALG_1:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_3.html#D1" target="_self" title="TOPALG_3:def.1">Def1</a></span>;<br/><a NAME="E11:41_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#K4" title="TOPALG_1:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_1_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E5:41_1_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="topalg_1.html#T62" title="TOPALG_1:th.62">TOPALG_1:62</a></span>;<br/><a NAME="E12:41_1_1_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_1_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="topalg_1.html#T47" title="TOPALG_1:th.47">TOPALG_1:47</a></span>;<br/><a NAME="E13:41_1_1_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_1_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="topalg_3.html#T28" target="_self" title="TOPALG_3:th.28">Th28</a></span>;<br/><span class="kw">hence </span><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#K4" title="TOPALG_1:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:41_1_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="topalg_1.html#T47" title="TOPALG_1:th.47">TOPALG_1:47</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_1_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E6:41_1_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="topalg_1.html#T62" title="TOPALG_1:th.62">TOPALG_1:62</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:41_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#NK6" title="TOPALG_1:NK.6">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#NK6" title="TOPALG_1:NK.6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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