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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="topalg_2.html#V1" target="_self">convex</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font>;<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#K1" target="_self">ConvexHomotopy</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>reconsider </b><font color="Maroon">x0</font> = 0, <font color="Maroon">x1</font> = 1 as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>, <font color="Maroon">b1</font> = <font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E3:8"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" target="_self">ConvexHomotopy</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>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="topalg_2.html#K1" target="_self">ConvexHomotopy</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 = 
<font color="Maroon">a1</font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">a1</font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/>

<a NAME="E5:8"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" target="_self">ConvexHomotopy</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>,<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="topalg_2.html#K1" target="_self">ConvexHomotopy</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"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<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/>

<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font>, <font color="Maroon">b1</font> = <font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" target="_self">ConvexHomotopy</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> <a href="hidden.html#R1">=</a> <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="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:8"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>;<br/>
<a NAME="E10:8"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>;<br/>
<a NAME="E11:8"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>;<br/>
<a NAME="E12:8"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><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#K1" target="_self">ConvexHomotopy</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="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
1 <a href="euclid.html#K18">*</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/>

<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font>, <font color="Maroon">b1</font> = <font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E15:8"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" target="_self">ConvexHomotopy</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> <a href="hidden.html#R1">=</a> <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="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="topalg_2.html#K1" target="_self">ConvexHomotopy</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="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
1 <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font>
<b>by </b><i>, , <a class="ref" href="euclid.html#T52">EUCLID:52</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/>


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