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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> 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="E1:20"/><i><font color="Green">E17</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="E2:20"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</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>5</sub></font>, <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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="funct_1.html#V3">constant</a>  <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>assume </b><a NAME="E4:20"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 ;<br/>

<a NAME="E5:20"/>
<font color="Maroon">c<sub>7</sub></font> <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#E2:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="borsuk_6.html#T92">BORSUK_6:92</a></i>;<br/>
<a NAME="E6:20"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <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#E1:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topalg_1.html#E4:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="borsuk_6.html#T83">BORSUK_6:83</a></i>;<br/>
<a NAME="E7:20"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <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#E1:20"><i><font color="Green">E17</font></i></a>, <a class="ref" href="borsuk_6.html#T88">BORSUK_6:88</a></i>;<br/>
<a NAME="E8:20"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E6:20"><i><font color="Green">E20</font></i></a>, <a class="ref" href="borsuk_6.html#T87">BORSUK_6:87</a></i>;<br/>
<a NAME="E9:20"/>
<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#E1:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topalg_1.html#E2:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="borsuk_6.html#T81">BORSUK_6:81</a></i>;<br/>
<b>hence </b><a NAME="E10:20"/>
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#E8:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="borsuk_6.html#T87">BORSUK_6:87</a></i>;<br/>


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