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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="borsuk_6.html#R1">are_connected</a> 
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="borsuk_6.html#R1">are_connected</a> 
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="borsuk_6.html#R1">are_connected</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>





<a NAME="E4:42"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p3">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:42"><i><font color="Green">E28</font></i></a>, <a class="txt" href="topalg_1.html#E2:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="topalg_1.html#E3:42"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topalg_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E5:42"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="borsuk_6.html#R1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="topalg_1.html#E3:42"><i><font color="Green">E30</font></i></a>, <a class="ref" href="borsuk_6.html#T46">BORSUK_6:46</a></i>;<br/>
<a NAME="E6:42"/>
 <a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>7</sub></font>, <a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E3:42"><i><font color="Green">E30</font></i></a>, <a class="ref" href="borsuk_2.html#T15">BORSUK_2:15</a></i>;<br/>
<a NAME="E7:42"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></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">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p4">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E3:42"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_1.html#E4:42"><i><font color="Green">E31</font></i></a>, <a class="txt" href="topalg_1.html#E5:42"><i><font color="Green">E32</font></i></a>, <a class="ref" href="borsuk_6.html#T83">BORSUK_6:83</a></i>;<br/>
<a NAME="E8:42"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p4">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:42"><i><font color="Green">E28</font></i></a>, <a class="txt" href="topalg_1.html#E2:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="topalg_1.html#E3:42"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topalg_1.html#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E9:42"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></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">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E7:42"><i><font color="Green">E33</font></i></a>, <a class="ref" href="borsuk_6.html#T87">BORSUK_6:87</a></i>;<br/>


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