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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">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">e</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="borsuk_6.html#R1">are_connected</a> 
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="borsuk_6.html#R1">are_connected</a> 
 <b>and </b><br/><a NAME="E3:32"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="borsuk_6.html#R1">are_connected</a> 
 <b>and </b><br/><a NAME="E4:32"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">d</font>,<font color="Maroon">e</font> <a href="borsuk_6.html#R1">are_connected</a> 
 ;<br/>

<b>let </b><font color="Maroon">A</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">B</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">C</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">D</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">d</font>,<font color="Maroon">e</font>;<br/>







<a NAME="E5:32"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">B</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">C</font>,<font color="Maroon">A</font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">C</font></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i>, , , <a class="ref" href="borsuk_6.html#T81">BORSUK_6:81</a></i>;<br/>
<a NAME="E6:32"/>
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="borsuk_6.html#R1">are_connected</a> 
 
<b>by </b><i>, , <a class="ref" href="borsuk_6.html#T46">BORSUK_6:46</a></i>;<br/>
<a NAME="E7:32"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="borsuk_6.html#R1">are_connected</a> 
 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T46">BORSUK_6:46</a></i>;<br/>
<a NAME="E8:32"/>
<font color="Maroon">D</font>,<font color="Maroon">D</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/>
<b>hence </b><a NAME="E9:32"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">B</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">C</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">D</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">D</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/>


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