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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">C</font> be  <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="c4">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>set </b><font color="Maroon" title="c5">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>reconsider </b><font color="Maroon" title="c6">g</font> =  <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="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">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#T48" target="_self" title="TOPALG_1:th.48">Th48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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="c8">P</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="E2:82_1"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon" title="c7">h</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="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c8">P</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="E3:82_1"/><i><font color="Green" title="E52">A2</font></i>: 
<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c8">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c3">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#T88" title="BORSUK_6:th.88">BORSUK_6:88</a></i>;<br/><b>thus </b><font color="Maroon" title="c7">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">g</font> = 
 <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="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c3">C</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E2:82_1"><i><font color="Green" title="E51">A1</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="c7">h</font>
<b>by </b><i><a class="txt" href="topalg_1.html#E2:82_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="topalg_1.html#E3:82_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>
;<br/><a NAME="E5:82_1"/><i><font color="Green" title="E53">A3</font></i>: 
<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c3">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">P</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="c6">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">h</font> = 
 <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="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">P</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E2:82_1"><i><font color="Green" title="E51">A1</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="c7">h</font>
<b>by </b><i><a class="txt" href="topalg_1.html#E2:82_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="topalg_1.html#E5:82_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:82"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <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> <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="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="ref" href="group_1.html#T10" title="GROUP_1:th.10">GROUP_1:10</a></i>;<br/>


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