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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>let </b><font color="Maroon">Q</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><b>that </b><br/><a NAME="E1:61"/><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> 
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 ;<br/>

<a NAME="E3:61"/>
for <font color="Olive">x</font> being  <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">X</font>,<font color="Maroon">x1</font></span>)</span> holds  <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_2.html#K3">.</a> <font color="Olive">x</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">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<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">X</font>,<font color="Maroon">x1</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">L</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E2:61_1"/><i><font color="Green">E47</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">X</font>,<font color="Maroon">x1</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>
<a NAME="E3:61_1"/>
<font color="Maroon">L</font>,<font color="Maroon">L</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T15">BORSUK_2:15</a></i>;<br/>
<a NAME="E4:61_1"/><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">L</font>,<font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">L</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i>, <a class="txt" href="topalg_1.html#E1:61"><i><font color="Green">E40</font></i></a>, <a class="ref" href="borsuk_6.html#T83">BORSUK_6:83</a></i>;<br/>
<a NAME="E5:61_1"/>
 <a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font>, <a href="borsuk_2.html#K2">-</a> <font color="Maroon">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i>, <a class="txt" href="topalg_1.html#E1:61"><i><font color="Green">E40</font></i></a>, <a class="ref" href="borsuk_6.html#T85">BORSUK_6:85</a></i>;<br/>
<a NAME="E6:61_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">L</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">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">L</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">Q</font></span>)</span> <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><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_2.html#K3">.</a> <font color="Maroon">x</font> = 
 <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">L</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 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">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">L</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">Q</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="topalg_1.html#T2" target="_self">Th2</a>, </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:61"/>
 <a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">P</font> <a href="funct_2.html#R2">=</a>  <a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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