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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="E1:28"/><i><font color="Green">E21</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="E2:28"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</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>assume </b><a NAME="E3:28"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 ;<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/>
<a NAME="E5:28"/><i><font color="Green">E24</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>by </b><i><a class="txt" href="topalg_1.html#E1:28"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topalg_1.html#E2:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="borsuk_6.html#T46">BORSUK_6:46</a></i>;<br/>
<a NAME="E6:28"/>
 <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#E2:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="borsuk_2.html#T15">BORSUK_2:15</a></i>;<br/>
<a NAME="E7:28"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</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>,<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#E2:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="topalg_1.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="topalg_1.html#E5:28"><i><font color="Green">E24</font></i></a>, <a class="ref" href="borsuk_6.html#T83">BORSUK_6:83</a></i>;<br/>
<a NAME="E8:28"/>
<font color="Maroon">c<sub>5</sub></font>,<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#E1:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="borsuk_2.html#T15">BORSUK_2:15</a></i>;<br/>
<a NAME="E9:28"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</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>,<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#E1:28"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topalg_1.html#E2:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="topalg_1.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E10:28"/><b>then </b><i><font color="Green">E26</font></i>: 
<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#E7:28"><i><font color="Green">E25</font></i></a>, <a class="ref" href="borsuk_6.html#T87">BORSUK_6:87</a></i>;<br/>
<a NAME="E11:28"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="borsuk_2.html#T15">BORSUK_2:15</a></i>;<br/>
<a NAME="E12:28"/><b>then </b>
<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>,<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:28"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topalg_1.html#E2:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="topalg_1.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E13:28"/>
<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> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E10:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="borsuk_6.html#T87">BORSUK_6:87</a></i>;<br/>


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