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

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   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/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">T</font><br/>  for <font color="Olive" title="b4">P</font>, <font color="Olive" title="b5">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font><br/>  for <font color="Olive" title="b6">s</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b4">P</font>,<font color="Olive" title="b5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b6">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">s</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b4">P</font>,<font color="Olive" title="b5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b6">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">s</font> &amp; (  for <font color="Olive" title="b7">t</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b4">P</font>,<font color="Olive" title="b5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b7">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b4">P</font>,<font color="Olive" title="b5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b7">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">x0</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <font color="Maroon" title="c4">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D14" onmouseover="rs('borsuk_1/D14')" onmouseout="rh()">BORSUK_1:def 14</a>, <a class="ref" href="borsuk_1.html#D15" onmouseover="rs('borsuk_1/D15')" onmouseout="rh()">BORSUK_1:def 15</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font><br/>  for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b5">s</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">s</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">s</font> &amp; (  for <font color="Olive" title="b6">t</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b6">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b6">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font><br/>  for <font color="Olive" title="b3">s</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">s</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">s</font> &amp; (  for <font color="Olive" title="b4">t</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">P</font>, <font color="Maroon" title="c9">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; (  for <font color="Olive" title="b2">t</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b2">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b2">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> ) ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c10">F</font> =  <a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>);<br/>
<span class="kw">let </span><font color="Maroon" title="c11">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> &amp; (  for <font color="Olive" title="b1">t</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> ) ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">a1</font> = <font color="Maroon" title="c8">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>, <font color="Maroon" title="c13">b1</font> = <font color="Maroon" title="c9">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T25" onmouseover="rs('pre_topc/T25')" onmouseout="rh()">PRE_TOPC:25</a></span>;<br/>
<a NAME="E3:8"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#D4" onmouseover="rs('borsuk_2/D4')" onmouseout="rh()">BORSUK_2:def 4</a></span>;<br/>
<a NAME="E4:8"/>
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c3">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">a1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_2.html#D2" target="_self" onmouseover="rs('topalg_2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) = 
<font color="Maroon" title="c12">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" onmouseover="rs('rlvect_1/D8')" onmouseout="rh()">RLVECT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c12">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" onmouseover="rs('rlvect_1/T10')" onmouseout="rh()">RLVECT_1:10</a></span>
<br/>.= 
<font color="Maroon" title="c8">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> &amp; (  for <font color="Olive" title="b1">t</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> ) ) )</span><br/>

<a NAME="E6:8"/>
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c4">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">a1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_2.html#D2" target="_self" onmouseover="rs('topalg_2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,1) = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" onmouseover="rs('rlvect_1/T10')" onmouseout="rh()">RLVECT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" onmouseover="rs('rlvect_1/D8')" onmouseout="rh()">RLVECT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c9">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c8">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x0</font>, <font color="Maroon" title="c15">b1</font> = <font color="Maroon" title="c9">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T25" onmouseover="rs('pre_topc/T25')" onmouseout="rh()">PRE_TOPC:25</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c16">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c16">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c16">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )</span><br/>

<a NAME="E9:8"/>
( <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c16">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">a1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">b1</font></span>)</span> &amp; <font color="Maroon" title="c9">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_2.html#D2" target="_self" onmouseover="rs('topalg_2/D2')" onmouseout="rh()">Def2</a>, <a class="ref" href="borsuk_2.html#D4" onmouseover="rs('borsuk_2/D4')" onmouseout="rh()">BORSUK_2:def 4</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c16">t</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">a1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="rlvect_1.html#T35" onmouseover="rs('rlvect_1/T35')" onmouseout="rh()">RLVECT_1:35</a></span>
<br/>.= 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">a1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_4.html#T1" onmouseover="rs('rlvect_4/T1')" onmouseout="rh()">RLVECT_4:1</a></span>
<br/>.= 
<font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="rlvect_1.html#D8" onmouseover="rs('rlvect_1/D8')" onmouseout="rh()">RLVECT_1:def 8</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c16">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font></span><br/>

<a NAME="E11:8"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c9">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#D4" onmouseover="rs('borsuk_2/D4')" onmouseout="rh()">BORSUK_2:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">a1</font> = <font color="Maroon" title="c8">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c18">b1</font> = <font color="Maroon" title="c9">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T25" onmouseover="rs('pre_topc/T25')" onmouseout="rh()">PRE_TOPC:25</a></span>;<br/>
<a NAME="E13:8"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#D4" onmouseover="rs('borsuk_2/D4')" onmouseout="rh()">BORSUK_2:def 4</a></span>;<br/>
<a NAME="E14:8"/>
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c16">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">a1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_2.html#D2" target="_self" onmouseover="rs('topalg_2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="topalg_2.html#K1" title="TOPALG_2:func.1">ConvexHomotopy</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c16">t</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">a1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E11:8"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="rlvect_1.html#T35" onmouseover="rs('rlvect_1/T35')" onmouseout="rh()">RLVECT_1:35</a></span>
<br/>.= 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">b1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E11:8"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="rlvect_4.html#T1" onmouseover="rs('rlvect_4/T1')" onmouseout="rh()">RLVECT_4:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="rlvect_1.html#D8" onmouseover="rs('rlvect_1/D8')" onmouseout="rh()">RLVECT_1:def 8</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
