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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">f</font> be  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font>;<br/><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</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="c9">P</font>, <font color="Maroon" title="c10">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>;<br/><b>let </b><font color="Maroon" title="c11">P1</font>, <font color="Maroon" title="c12">Q1</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font>;<br/><b>let </b><font color="Maroon" title="c13">F</font> be    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c9">P</font>,<font color="Maroon" title="c10">Q</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:35"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c9">P</font>,<font color="Maroon" title="c10">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c11">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">P</font>
 <b>and </b><br/><a NAME="E3:35"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c12">Q1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">Q</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c14">G</font> = <font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">F</font>;<br/>
<b>thus </b><a NAME="E4:35"/>
<font color="Maroon" title="c11">P1</font>,<font color="Maroon" title="c12">Q1</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:35"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="topalg_3.html#E2:35"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="topalg_3.html#E3:35"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="topalg_3.html#T28" target="_self" title="TOPALG_3:th.28">Th28</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a><br/>

<a NAME="E5:35"/>
<font color="Maroon" title="c13">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E1:35"><i><font color="Green" title="E13">A1</font></i></a>, <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="E6:35"/>
<font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>

<b>let </b><font color="Maroon" title="c15">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c2">j0</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binop_1.html#T30" title="BINOP_1:th.30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">s</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E1:35"><i><font color="Green" title="E13">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="c11">P1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">s</font>
<b>by </b><i><a class="txt" href="topalg_3.html#E2:35"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">s</font>,1 = 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c3">j1</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binop_1.html#T30" title="BINOP_1:th.30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">s</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E1:35"><i><font color="Green" title="E13">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="c12">Q1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">s</font>
<b>by </b><i><a class="txt" href="topalg_3.html#E3:35"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">F</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="c15">s</font> = 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">j0</font>,<font color="Maroon" title="c15">s</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binop_1.html#T30" title="BINOP_1:th.30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>
<b>by </b><i><a class="txt" href="topalg_3.html#E1:35"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c15">s</font> = 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j1</font>,<font color="Maroon" title="c15">s</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binop_1.html#T30" title="BINOP_1:th.30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="txt" href="topalg_3.html#E1:35"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>
;<br/>


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