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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c6">A</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/><b>let </b><font color="Maroon" title="c7">B</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>;<br/><b>let </b><font color="Maroon" title="c8">C</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>;<br/>





<a NAME="E4:40"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c7">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c7">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T15" title="BORSUK_2:th.15">BORSUK_2:15</a></i>;<br/>
<a NAME="E5:40"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c8">C</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="borsuk_2.html#T15" title="BORSUK_2:th.15">BORSUK_2:15</a></i>;<br/>
<a NAME="E6:40"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c8">C</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="topalg_1.html#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="topalg_1.html#T26" target="_self" title="TOPALG_1:th.26">Th26</a></i>;<br/>
<a NAME="E7:40"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p3">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span></span>)</span>,<font color="Maroon" title="c8">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span></span>)</span> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="topalg_1.html#E4:40"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="borsuk_6.html#T83" title="BORSUK_6:th.83">BORSUK_6:83</a></i>;<br/>
<a NAME="E8:40"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p4">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p3">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span></span>)</span> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="topalg_1.html#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="borsuk_6.html#T81" title="BORSUK_6:th.81">BORSUK_6:81</a></i>;<br/>
<a NAME="E9:40"/><b>then </b><i><font color="Green" title="E33">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p4">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c8">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span></span>)</span> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E7:40"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="borsuk_6.html#T87" title="BORSUK_6:th.87">BORSUK_6:87</a></i>;<br/>
<a NAME="E10:40"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c8">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="topalg_1.html#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="topalg_1.html#E5:40"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="topalg_1.html#T20" target="_self" title="TOPALG_1:th.20">Th20</a></i>;<br/>
<a NAME="E11:40"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c8">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span></span>)</span> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="topalg_1.html#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="borsuk_6.html#T81" title="BORSUK_6:th.81">BORSUK_6:81</a></i>;<br/>
<a NAME="E12:40"/><b>then </b>
<font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c8">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span></span>)</span> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E10:40"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="borsuk_6.html#T87" title="BORSUK_6:th.87">BORSUK_6:87</a></i>;<br/>
<b>hence </b><a NAME="E13:40"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p4">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c8">C</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E9:40"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="borsuk_6.html#T87" title="BORSUK_6:th.87">BORSUK_6:87</a></i>;<br/>


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