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<b>let </b><font color="Maroon">S</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">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">s1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">t1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">t2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">s1</font>,<font color="Maroon">s2</font> <a href="borsuk_6.html#R1">are_connected</a> 
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="borsuk_6.html#R1">are_connected</a> 
 ;<br/>

<b>let </b><font color="Maroon">L1</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">s1</font>,<font color="Maroon">s2</font>;<br/><b>let </b><font color="Maroon">L2</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">t1</font>,<font color="Maroon">t2</font>;<br/>



<a NAME="E3:20"/>
( <font color="Maroon">L1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">L2</font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i>, , <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/>
<a NAME="E4:20"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="yellow12.html#T41">YELLOW12:41</a></i>;<br/>
<a NAME="E5:20"/><i><font color="Green">E43</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E44</font></i>: <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j0</font> = 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j0</font></span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#T69">FUNCT_3:69</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s1</font>,<span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j0</font></span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
<b>by </b><i>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">t1</font></span><a href="borsuk_1.html#K7">]</a></span>
<b>by </b><i>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
;<br/>
<i><font color="Green">E47</font></i>: <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font> = 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font></span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_3.html#T69">FUNCT_3:69</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s2</font>,<span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font></span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
<b>by </b><i>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">t2</font></span><a href="borsuk_1.html#K7">]</a></span>
<b>by </b><i>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
;<br/>
<a NAME="E8:20"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">t1</font></span><a href="borsuk_1.html#K7">]</a></span>,<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">t2</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="borsuk_6.html#R1">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="E9:20"/>
<span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> is    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">t1</font></span><a href="borsuk_1.html#K7">]</a></span>,<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">t2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, , , <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/>


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