<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="topalg_2.html#V1" title="TOPALG_2:attr.1">convex</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font>;<br/><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">T</font>;<br/><b>let </b><font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c7">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>;<br/>







<b>set </b><font color="Maroon" title="c8">F</font> =  <a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</font>;<br/>
<b>let </b><font color="Maroon" title="c9">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">x0</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c11">x1</font> = 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">a1</font> = <font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s</font>, <font color="Maroon" title="c13">b1</font> = <font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></i>;<br/>
<a NAME="E3:8"/>
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c10">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">x0</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c12">a1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x0</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c13">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_2.html#D2" target="_self" title="TOPALG_2:def.2">Def2</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c12">a1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c13">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c12">a1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
;<br/><br/>

<a NAME="E5:8"/>
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">x1</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c12">a1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c13">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_2.html#D2" target="_self" title="TOPALG_2:def.2">Def2</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">s</font>,1 = 
<span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c13">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c13">b1</font>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
;<br/><br/>

<b>let </b><font color="Maroon" title="c14">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c15">a1</font> = <font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x0</font>, <font color="Maroon" title="c16">b1</font> = <font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green" title="E10">A1</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</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">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">t</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c15">a1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c16">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_2.html#D2" target="_self" title="TOPALG_2:def.2">Def2</a></i>;<br/>
<a NAME="E9:8"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></i>;<br/>
<a NAME="E10:8"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></i>;<br/>
<a NAME="E11:8"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></i>;<br/>
<a NAME="E12:8"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</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">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c15">a1</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c15">a1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c15">a1</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E8:8"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="topalg_2.html#E9:8"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="topalg_2.html#E10:8"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="euclid.html#T54" title="EUCLID:th.54">EUCLID:54</a></i>
<br/>.= 
1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c15">a1</font>
<b>by </b><i><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="topalg_2.html#E9:8"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
;<br/>

<b>reconsider </b><font color="Maroon" title="c17">a1</font> = <font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c18">b1</font> = <font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></i>;<br/>
<a NAME="E15:8"/>
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">t</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c17">a1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c18">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_2.html#D2" target="_self" title="TOPALG_2:def.2">Def2</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c14">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c17">a1</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c18">b1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c17">a1</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E11:8"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="topalg_2.html#E12:8"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="euclid.html#T54" title="EUCLID:th.54">EUCLID:54</a></i>
<br/>.= 
1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c18">b1</font>
<b>by </b><i><a class="txt" href="topalg_2.html#E11:8"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="topalg_2.html#E12:8"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>
<br/>.= 
<font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="txt" href="topalg_2.html#E12:8"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
;<br/><br/>


</div>
