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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>, <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">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>, <font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</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>, <font color="Maroon">Q</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">P1</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>, <font color="Maroon">Q1</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">P</font>,<font color="Maroon">Q</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">P1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Q1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Q</font>
 ;<br/>

<b>set </b><font color="Maroon">G</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font>;<br/>
<b>thus </b><a NAME="E4:34"/>
<font color="Maroon">P1</font>,<font color="Maroon">Q1</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i>, , , </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a><br/>

<a NAME="E5:34"/>
<font color="Maroon">F</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="E6:34"/>
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>

<b>let </b><font color="Maroon">s</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"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,0 = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">j0</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binop_1.html#T30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon">P1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,1 = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">j1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binop_1.html#T30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon">Q1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>

<b>let </b><font color="Maroon">t</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"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j0</font>,<font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binop_1.html#T30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j1</font>,<font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binop_1.html#T30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
;<br/>


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