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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="topalg_2.html#V3" target="_self">convex</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topalg_2.html#K3" target="_self">R^1</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</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>set </b><font color="Maroon">F</font> =  <a href="topalg_2.html#K4" target="_self">R1Homotopy</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font>;<br/>
<b>let </b><font color="Maroon">s</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_2.html#K4" target="_self">R1Homotopy</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>

;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_2.html#K4" target="_self">R1Homotopy</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a></i>
<br/>.= 
<font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>

;<br/>

<b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E3:35"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K21">0[01]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>;<br/>
<a NAME="E4:35"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K21">0[01]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>;<br/>
<a NAME="E5:35"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K22">1[01]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>;<br/>
<a NAME="E6:35"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K22">1[01]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="topalg_2.html#K4" target="_self">R1Homotopy</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, </i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_2.html#K4" target="_self">R1Homotopy</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K22">1[01]</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K22">1[01]</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, </i>
;<br/>


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