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<b>let </b><font color="Maroon">T</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">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">Q</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><b>let </b><font color="Maroon">R</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:102"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 <b>and </b><br/><a NAME="E2:102"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 <b>and </b><br/><a NAME="E3:102"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 ;<br/>

<a NAME="E4:102"/>
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T25" target="_self">Th25</a>, , </i>;<br/>
<a NAME="E5:102"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:102"/>
 <a href="borsuk_6.html#K2" target="_self">RePar</a> <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">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a>  <a href="funct_2.html#R2">=</a> <font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T25" target="_self">Th25</a>, , , <a class="ref" href="borsuk_6.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E7:102"/>
<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> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font>,<font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i>, <a class="ref" href="borsuk_6.html#T7" target="_self">Th7</a>, <a class="ref" href="borsuk_6.html#T12" target="_self">Th12</a></i>;<br/>


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