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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>



<b>set </b><font color="Maroon">X</font> =  <a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">S</font>,<font color="Maroon">s</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<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>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<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>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">s</font>, <font color="Maroon">Q</font> be   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">s</font>;<br/>



<b>assume </b><a NAME="E1:58"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">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">S</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">P</font> &amp; <font color="Maroon">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">S</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">Q</font> )
 ;<br/>

<b>consider </b><font color="Maroon">P1</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">s</font>, <font color="Maroon">Q1</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">s</font><b> such that </b><br/><a NAME="E3:58"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">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">S</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">P1</font> &amp; <font color="Maroon">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">S</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">Q1</font> )
 <b>and </b><br/><a NAME="E4:58"/><i><font color="Green">E47</font></i>: 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<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> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">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">S</font>,<font color="Maroon">s</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/>
<a NAME="E5:58"/>
( <font color="Maroon">P</font>,<font color="Maroon">P1</font> <a href="borsuk_2.html#R3">are_homotopic</a>  &amp; <font color="Maroon">Q</font>,<font color="Maroon">Q1</font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:58"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font>,<font color="Maroon">P1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q1</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>thus </b><a NAME="E7:58"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">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">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="topalg_1.html#E1:58"><i><font color="Green">E40</font></i></a>, </i>;<br/>


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