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

<b>let </b><font color="Maroon">T</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">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E4:53"/>
( <font color="Maroon">P</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<font color="Maroon">T</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="borsuk_6.html#T5" target="_self">Th5</a>, <a class="ref" href="borsuk_6.html#T1" target="_self">Th1</a>, <a class="txt" href="borsuk_6.html#E2"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<a NAME="E5:53"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="borsuk_2.html#R2">are_connected</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:53"/>
(  <a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/>


</div>
