<?xml version="1.0"?>
<div class="add">

<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>, <font color="Maroon" title="c6">e</font>, <font color="Maroon" title="c7">f</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="E1:44"/><i><font color="Green" title="E29">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="E2:44"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c3">b</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="E3:44"/><i><font color="Green" title="E31">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> 
 <b>and </b><br/><a NAME="E4:44"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">e</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <b>and </b><br/><a NAME="E5:44"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">f</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 ;<br/>

<a NAME="E6:44"/><i><font color="Green" title="E34">A6</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>by </b><i><a class="txt" href="topalg_1.html#E1:44"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="topalg_1.html#E2:44"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="borsuk_6.html#T46" title="BORSUK_6:th.46">BORSUK_6:46</a></i>;<br/>
<a NAME="E7:44"/><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">e</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E3:44"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="topalg_1.html#E4:44"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="borsuk_6.html#T46" title="BORSUK_6:th.46">BORSUK_6:46</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">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="c9">B</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>;<br/><b>let </b><font color="Maroon" title="c10">C</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="c11">D</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">e</font>;<br/><b>let </b><font color="Maroon" title="c12">E</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c4">c</font>;<br/>









<b>consider </b><font color="Maroon" title="c13">X</font> being  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <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="c4">c</font>;<br/>
<a NAME="E9:44"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E5:44"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="topalg_1.html#E6:44"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="borsuk_6.html#T46" title="BORSUK_6:th.46">BORSUK_6:46</a></i>;<br/>
<a NAME="E10:44"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <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="c12">E</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c12">E</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="c10">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</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="c10">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</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#E6:44"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="topalg_1.html#E7:44"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="topalg_1.html#T38" target="_self" title="TOPALG_1:th.38">Th38</a></i>;<br/>
<a NAME="E11:44"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</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="c10">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">C</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</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#E1:44"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="topalg_1.html#E2:44"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="topalg_1.html#E3:44"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="topalg_1.html#E4:44"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="topalg_1.html#T36" target="_self" title="TOPALG_1:th.36">Th36</a></i>;<br/>
<a NAME="E12:44"/><b>then </b><i><font color="Green" title="E38">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">C</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <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="c12">E</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c12">E</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="c10">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</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#E10:44"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="borsuk_6.html#T87" title="BORSUK_6:th.87">BORSUK_6:87</a></i>;<br/>
<a NAME="E13:44"/><i><font color="Green" title="E39">A11</font></i>: 
<font color="Maroon" title="c12">E</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">C</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</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#E3:44"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="topalg_1.html#E4:44"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="topalg_1.html#E9:44"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="borsuk_6.html#T81" title="BORSUK_6:th.81">BORSUK_6:81</a></i>;<br/>
<a NAME="E14:44"/><i><font color="Green" title="E40">A12</font></i>: 
<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c6">e</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E7:44"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="topalg_1.html#E9:44"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="borsuk_6.html#T46" title="BORSUK_6:th.46">BORSUK_6:46</a></i>;<br/>
<a NAME="E15:44"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">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="c12">E</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">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="c12">E</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#E5:44"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="borsuk_2.html#T15" title="BORSUK_2:th.15">BORSUK_2:15</a></i>;<br/>
<a NAME="E16:44"/><b>then </b><i><font color="Green" title="E41">A13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <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="c12">E</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <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="c12">E</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">C</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</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#E5:44"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="topalg_1.html#E13:44"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="topalg_1.html#E14:44"><i><font color="Green" title="E40">A12</font></i></a>, <a class="ref" href="borsuk_6.html#T83" title="BORSUK_6:th.83">BORSUK_6:83</a></i>;<br/>
<a NAME="E17:44"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <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="c12">E</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <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="c12">E</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c12">E</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="c10">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</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#E5:44"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="topalg_1.html#E7:44"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="topalg_1.html#E9:44"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="borsuk_6.html#T81" title="BORSUK_6:th.81">BORSUK_6:81</a></i>;<br/>
<a NAME="E18:44"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <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="c12">E</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c12">E</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="c10">C</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <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="c12">E</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">C</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</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#E16:44"><i><font color="Green" title="E41">A13</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="E19:44"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">C</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <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="c12">E</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c10">C</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">D</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#E12:44"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="borsuk_6.html#T87" title="BORSUK_6:th.87">BORSUK_6:87</a></i>;<br/>


</div>
