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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>let </b><font color="Maroon">l1</font> be    <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>;<br/><b>let </b><font color="Maroon">l2</font> be    <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>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">l1</font>,<font color="Maroon">l2</font>;<br/>

















<b>assume </b><a NAME="E1:26"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">l1</font>,<font color="Maroon">l2</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 ;<br/>

<a NAME="E2:26"/><b>then </b>
<font color="Maroon">H</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:26"/>
 <a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/>

<a NAME="E4:26"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="borsuk_1.html#K20">I[01]</a> ,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green">E41</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> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:26"/><i><font color="Green">E43</font></i>: 
 <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>thus </b><span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,0 = 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">j0</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">j0</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">l1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,1 = 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">j1</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">j1</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">l2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>
;<br/>

<b>let </b><font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">j0</font>,<font color="Maroon">b</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j0</font>,<font color="Maroon">b</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 

<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> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon">s1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">j1</font>,<font color="Maroon">b</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j1</font>,<font color="Maroon">b</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 

<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> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon">s2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>


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