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<b>let </b><font color="Maroon" title="c4">S</font>, <font color="Maroon" title="c5">T</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="c6">s1</font>, <font color="Maroon" title="c7">s2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">S</font>;<br/><b>let </b><font color="Maroon" title="c8">t1</font>, <font color="Maroon" title="c9">t2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font>;<br/><b>let </b><font color="Maroon" title="c10">l1</font>, <font color="Maroon" title="c11">l2</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>,<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>;<br/><b>let </b><font color="Maroon" title="c12">H</font> be    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c10">l1</font>,<font color="Maroon" title="c11">l2</font>;<br/>

















<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c10">l1</font>,<font color="Maroon" title="c11">l2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 ;<br/>

<a NAME="E2:26"/><b>then </b>
<font color="Maroon" title="c12">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:26"/>
 <a href="topalg_4.html#K4" title="TOPALG_4:func.4">pr1</a> <font color="Maroon" title="c12">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topalg_4.html#T9" target="_self" title="TOPALG_4:th.9">Th9</a></i>;<br/>

<b>let </b><font color="Maroon" title="c13">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ;<br/>

<a NAME="E4:26"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green" title="E22">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:26"/><i><font color="Green" title="E23">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="topalg_4.html#K4" title="TOPALG_4:func.4">pr1</a> <font color="Maroon" title="c12">H</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">a</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c2">j0</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E4:26"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c2">j0</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">l1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:26"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_4.html#K4" title="TOPALG_4:func.4">pr1</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font>
<b>by </b><i><a class="txt" href="topalg_4.html#E5:26"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_4.html#K4" title="TOPALG_4:func.4">pr1</a> <font color="Maroon" title="c12">H</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">a</font>,1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c3">j1</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E4:26"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c3">j1</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">l2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:26"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_4.html#K4" title="TOPALG_4:func.4">pr1</a> <font color="Maroon" title="c11">l2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font>
<b>by </b><i><a class="txt" href="topalg_4.html#E6:26"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>
;<br/>

<b>let </b><font color="Maroon" title="c14">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_4.html#K4" title="TOPALG_4:func.4">pr1</a> <font color="Maroon" title="c12">H</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c2">j0</font>,<font color="Maroon" title="c14">b</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E4:26"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">j0</font>,<font color="Maroon" title="c14">b</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 

<br/>.= 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:26"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c6">s1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_4.html#K4" title="TOPALG_4:func.4">pr1</a> <font color="Maroon" title="c12">H</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c14">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c3">j1</font>,<font color="Maroon" title="c14">b</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E4:26"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j1</font>,<font color="Maroon" title="c14">b</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 

<br/>.= 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:26"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c7">s2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/>


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