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<b>let </b><font color="Maroon">X</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">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">x0</font>,<font color="Maroon">x1</font>;<br/>







<b>assume </b><a NAME="E1:65"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x0</font>,<font color="Maroon">x1</font> <a href="borsuk_6.html#R1">are_connected</a> 
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">P</font>;<br/>
<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:65"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">P</font></span>)</span>
 <b>and </b><br/><a NAME="E3:65"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">P</font></span>)</span>
 <b>and </b><br/><a NAME="E4:65"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E6:65"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</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">x1</font></span>)</span>,<font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E8:65"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</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">x1</font></span>)</span>,<font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:65"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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">x0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">A</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_1.html#T2" target="_self">Th2</a>, , </i>;<br/>
<a NAME="E10:65"/>
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</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">x0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">B</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_1.html#T2" target="_self">Th2</a>, <a class="txt" href="topalg_1.html#E2:65"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E11:65"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">A</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">B</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i>, <a class="txt" href="topalg_1.html#E3:65"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<a NAME="E12:65"/><b>then </b>
<font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">A</font>,<font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">B</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="E13:65"/><b>then </b>
<font color="Maroon">A</font>,<font color="Maroon">B</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/>
<b>hence </b><a NAME="E14:65"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="topalg_1.html#E2:65"><i><font color="Green">E46</font></i></a>, </i>;<br/>


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