<?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>set </b><font color="Maroon" title="c9">E</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>;<br/>
<b>set </b><font color="Maroon" title="c10">PI</font> = <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c6">P</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c11">QI</font> = <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c7">Q</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>;<br/>
<a NAME="E1:7"/><i><font color="Green" title="E9">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">G</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> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<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="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<a NAME="E3:7"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c14">b1</font> = <font color="Maroon" title="c5">b</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="txt" href="topalg_2.html#E1:7"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E5:7"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D3" title="BORSUK_2:def.3">BORSUK_2:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">P1</font> = <font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c16">Q1</font> = <font color="Maroon" title="c7">Q</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c14">b1</font> <b>by </b><i><a class="ref" href="topalg_2.html#T1" target="_self" title="TOPALG_2:th.1">Th1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">P1I</font> = <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c15">P1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c18">Q1I</font> = <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c16">Q1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>;<br/>
<a NAME="E7:7"/><i><font color="Green" title="E10">A2</font></i>: 
( <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c15">P1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c16">Q1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T12" title="BORSUK_2:th.12">BORSUK_2:12</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> $2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> = $2 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c19">Pa</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> such that </b><br/><a NAME="E9:7"/><i><font color="Green" title="E11">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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><br/> for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c19">Pa</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c20">Pb</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> such that </b><br/><a NAME="E11:7"/><i><font color="Green" title="E12">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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><br/> for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c20">Pb</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E12:7"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><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>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">Pa</font> = <font color="Maroon" title="c19">Pa</font>, <font color="Maroon" title="c22">Pb</font> = <font color="Maroon" title="c20">Pb</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><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>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<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> ;<br/>
<a NAME="E14:7"/>
( <font color="Maroon" title="c21">Pa</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c22">Pb</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="topalg_2.html#E9:7"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="topalg_2.html#E11:7"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="topalg_1.html#T18" title="TOPALG_1:th.18">TOPALG_1:18</a>, <a class="ref" href="topalg_1.html#T19" title="TOPALG_1:th.19">TOPALG_1:19</a></i>;<br/>
<a NAME="E15:7"/><b>then </b><i><font color="Green" title="E13">A5</font></i>: 
( <font color="Maroon" title="c21">Pa</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c15">P1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c22">Pb</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c16">Q1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="topalg_2.html#E7:7"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E16:7"/><i><font color="Green" title="E14">A6</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Pa</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c15">P1</font>,<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">Pb</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c16">Q1</font>,<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>;<br/>

<b>consider </b><font color="Maroon" title="c24">s</font>, <font color="Maroon" title="c25">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> <b> such that </b><br/><a NAME="E2:7_1"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c24">s</font>,<font color="Maroon" title="c25">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<a NAME="E3:7_1"/><i><font color="Green" title="E16">A8</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:7_1"/><i><font color="Green" title="E17">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:7_1"/><i><font color="Green" title="E18">A10</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">t</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<i><font color="Green" title="E19">A11</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c21">Pa</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">p</font> = 
<font color="Maroon" title="c21">Pa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c24">s</font>,<font color="Maroon" title="c25">t</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E2:7_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c21">Pa</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">s</font></span>)</span>,<font color="Maroon" title="c25">t</font>
<b>by </b><i><a class="txt" href="topalg_2.html#E3:7_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="topalg_2.html#E5:7_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="topalg_2.html#E6"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c25">t</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">P1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">s</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E9:7"><i><font color="Green" title="E11">A3</font></i></a></i>
;<br/>
<i><font color="Green" title="E20">A12</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c22">Pb</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c7">Q</font>,<span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">p</font> = 
<font color="Maroon" title="c22">Pb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c7">Q</font>,<span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c24">s</font>,<font color="Maroon" title="c25">t</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E2:7_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c22">Pb</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">s</font></span>)</span>,<font color="Maroon" title="c25">t</font>
<b>by </b><i><a class="txt" href="topalg_2.html#E4:7_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="topalg_2.html#E5:7_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="topalg_2.html#E6"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c25">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">Q1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">s</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E11:7"><i><font color="Green" title="E12">A4</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c26">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="c24">s</font>, <font color="Maroon" title="c27">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="c24">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="E9:7_1"/>
<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="c24">s</font>,<font color="Maroon" title="c25">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="c25">t</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c26">a1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c27">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><a NAME="E10:7_1"/>
<font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Pa</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c15">P1</font>,<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">Pb</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c16">Q1</font>,<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_2.html#E2:7_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="topalg_2.html#E6:7_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="topalg_2.html#E7:7_1"><i><font color="Green" title="E20">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c23">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<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> such that </b><br/><a NAME="E18:7"/><i><font color="Green" title="E15">A13</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon" title="c23">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Pa</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c15">P1</font>,<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">Pb</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c16">Q1</font>,<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">r</font></span>)</span>
 <b>and </b><br/><a NAME="E19:7"/><i><font color="Green" title="E16">A14</font></i>: 
<font color="Maroon" title="c23">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_2.html#E15:7"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="jgraph_6.html#T20" title="JGRAPH_6:th.20">JGRAPH_6:20</a></i>;<br/>
<a NAME="E20:7"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>;<br/>

<b>thus </b><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Pa</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c15">P1</font>,<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">Pb</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c16">Q1</font>,<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">p</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E16:7"><i><font color="Green" title="E14">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c23">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">p</font>
<b>by </b><i><a class="txt" href="topalg_2.html#E18:7"><i><font color="Green" title="E15">A13</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E21:7"/><b>then </b>
<font color="Maroon" title="c12">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c23">g</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E22:7"/>
 <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> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_2.html#E19:7"><i><font color="Green" title="E16">A14</font></i></a>, <a class="ref" href="topmetr.html#T8" title="TOPMETR:th.8">TOPMETR:8</a></i>;<br/>


</div>
