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<b>let </b><font color="Maroon">F</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:54_1_2"/><i><font color="Green">E51</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>
 <b>and </b><br/><a NAME="E2:54_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>  ex <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> st <br/>( <font color="Olive">x</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Olive">P</font> &amp; <font color="Olive">y</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Olive">Q</font> &amp; the <a href="group_1.html#U1">mult</a> of <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Olive">Q</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:54_1_2"/><i><font color="Green">E81</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>
 <b>and </b><br/><a NAME="E4:54_1_2"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  ex <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> st <br/>( <font color="Olive">x</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Olive">P</font> &amp; <font color="Olive">y</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Olive">Q</font> &amp; the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Olive">Q</font></span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon">f</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">g</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font>;<br/>
<a NAME="E5:54_1_2"/>
for <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> holds  the <a href="group_1.html#U1">mult</a> of <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">c</font>,<font color="Olive">d</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Olive">c</font>,<font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>;<br/>

<b>consider </b><font color="Maroon">P1</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font>, <font color="Maroon">Q1</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font><b> such that </b><br/><a NAME="E2:54_1_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Maroon">P1</font>
 <b>and </b><br/><a NAME="E3:54_1_2_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">d</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Maroon">Q1</font>
 <b>and </b><br/><a NAME="E4:54_1_2_1"/><i><font color="Green">E95</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">d</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q1</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">P2</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font>, <font color="Maroon">Q2</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font><b> such that </b><br/><a NAME="E6:54_1_2_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Maroon">P2</font>
 <b>and </b><br/><a NAME="E7:54_1_2_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">d</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Maroon">Q2</font>
 <b>and </b><br/><a NAME="E8:54_1_2_1"/><i><font color="Green">E99</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">d</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" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q2</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E9:54_1_2_1"/>
( <font color="Maroon">P1</font>,<font color="Maroon">P2</font> <a href="borsuk_2.html#R3">are_homotopic</a>  &amp; <font color="Maroon">Q1</font>,<font color="Maroon">Q2</font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 
<b>by </b><i>, , <a class="ref" href="topalg_1.html#T2" target="_self">Th2</a>, , </i>;<br/>
<a NAME="E10:54_1_2_1"/><b>then </b>
<font color="Maroon">P1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q1</font>,<font color="Maroon">P2</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q2</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T83">BORSUK_6:83</a></i>;<br/>
<b>hence </b><a NAME="E11:54_1_2_1"/>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:54_1_2"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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