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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/>

<b>set </b><font color="Maroon" title="c2">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/>
<b>set </b><font color="Maroon" title="c3">p</font> =  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c4">U</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> </span>}</span> ;<br/>
<a NAME="E1:36"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> </span>}</span> 
 ;<br/>

<a NAME="E2:36_1"/><b>then </b>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> )
 
;<br/>
<b>hence </b><a NAME="E3:36_1"/>
<font color="Maroon" title="c5">x</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">U</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> </span>}</span>  as   <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> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">n0</font> =  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">H</font> =  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c8">q</font> =  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">S</font> =  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E3:36"/><i><font color="Green" title="E33">A1</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green" title="E34">A2</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E5:36"/><i><font color="Green" title="E35">A3</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E6:36"/><i><font color="Green" title="E36">A4</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green" title="E37">A5</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E8:36"/><i><font color="Green" title="E38">A6</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E9:36"/><i><font color="Green" title="E39">A7</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E10:36"/><i><font color="Green" title="E40">A8</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E11:36"/><i><font color="Green" title="E41">A9</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 
<b>by </b><i><a class="txt" href="jordan22.html#E5:36"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="jordan22.html#E7:36"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/>
<a NAME="E12:36"/><i><font color="Green" title="E42">A10</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="jordan21.html#T48" title="JORDAN21:th.48">JORDAN21:48</a></i>;<br/>
<a NAME="E13:36"/><i><font color="Green" title="E43">A11</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:36_2"/><i><font color="Green" title="E44">A12</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1g.html#T41" title="JORDAN1G:th.41">JORDAN1G:41</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="ref" href="jordan21.html#T28" title="JORDAN21:th.28">JORDAN21:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="txt" href="jordan22.html#E1:36_2"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="jordan21.html#T29" title="JORDAN21:th.29">JORDAN21:29</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E14:36"/><i><font color="Green" title="E44">A13</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>set </b><font color="Maroon" title="c11">f</font> =  <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font>;<br/>
<b>set </b><font color="Maroon" title="c12">s</font> =  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:36_3"/><i><font color="Green" title="E45">A14</font></i>: 
 <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan21.html#T61" title="JORDAN21:th.61">JORDAN21:61</a></i>;<br/>
<b>assume </b><a NAME="E2:36_3"/><i><font color="Green" title="E46">A15</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">n</font>
 ;<br/>

<a NAME="E3:36_3"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:36_3"/><b>then </b>
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:36_3"/><b>then </b>
 <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1a.html#T90" title="JORDAN1A:th.90">JORDAN1A:90</a></i>;<br/>
<a NAME="E6:36_3"/><b>then </b><i><font color="Green" title="E47">A16</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E1:36_3"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:36_3"/><i><font color="Green" title="E48">A17</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="jordan22.html#E13:36"><i><font color="Green" title="E43">A11</font></i></a></i>;<br/>
<a NAME="E8:36_3"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E2:36_3"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="jordan22.html#T26" target="_self" title="JORDAN22:th.26">Th26</a></i>;<br/>
<a NAME="E9:36_3"/><b>then </b><i><font color="Green" title="E49">A18</font></i>: 
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E5:36"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="jordan22.html#E6:36"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="jordan22.html#E10:36"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="jordan22.html#E6:36_3"><i><font color="Green" title="E47">A16</font></i></a>, <a class="txt" href="jordan22.html#E7:36_3"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="goboard7.html#T8" title="GOBOARD7:th.8">GOBOARD7:8</a></i>;<br/>
<a NAME="E10:36_3"/>
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U</font>
 
<b>by </b><i><a class="txt" href="jordan22.html#E2:36_3"><i><font color="Green" title="E46">A15</font></i></a></i>;<br/>
<a NAME="E11:36_3"/><b>then </b><i><font color="Green" title="E50">A19</font></i>: 
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font>
 
<b>by </b><i><a class="txt" href="jordan22.html#E9:36_3"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:36_3"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>;<br/>
<b>hence </b><a NAME="E13:36_3"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36_3"><i><font color="Green" title="E50">A19</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:36"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T98" title="TOPREAL6:th.98">TOPREAL6:98</a></i>;<br/>
<a NAME="E16:36"/><b>then </b>
<a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
<b>by </b><i><a class="ref" href="jct_misc.html#T23" title="JCT_MISC:th.23">JCT_MISC:23</a></i>;<br/>
<a NAME="E17:36"/><b>then </b><i><font color="Green" title="E45">A20</font></i>: 
<a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E18:36"/><i><font color="Green" title="E46">A21</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan21.html#T61" title="JORDAN21:th.61">JORDAN21:61</a></i>;<br/>
<a NAME="E19:36"/><i><font color="Green" title="E47">A22</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan22.html#T26" target="_self" title="JORDAN22:th.26">Th26</a></i>;<br/>
<a NAME="E20:36"/><i><font color="Green" title="E48">A23</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E14:36"><i><font color="Green" title="E44">A13</font></i></a></i>;<br/>
<a NAME="E21:36"/><b>then </b>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E17:36"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E22:36"/><b>then </b><i><font color="Green" title="E49">A24</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E18:36"><i><font color="Green" title="E46">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_4"/>
(  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_1"><b>suppose </b></a><a NAME="E1:36_4_1"/><i><font color="Green" title="E50">A25</font></i>: 
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><div><i><font color="Green" title="E51">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4_1_1"/><i><font color="Green" title="E52">A27</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_4_1_1_1"/>
( <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan22.html#E1:36_4_1_1"><i><font color="Green" title="E52">A27</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_4_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_4_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan22.html#E1:36_4_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_4_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:36_4_1_1_1_2"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:36_4_1_1_1_2"/><i><font color="Green" title="E53">A28</font></i>: 
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span>
 <b>and </b><br/><a NAME="E5:36_4_1_1_1_2"/><i><font color="Green" title="E54">A29</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E17:36"><i><font color="Green" title="E45">A20</font></i></a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">t</font> being   <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> 2</span>)</span><b> such that </b><br/><a NAME="E7:36_4_1_1_1_2"/><i><font color="Green" title="E55">A30</font></i>: 
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font>
 <b>and </b><br/><a NAME="E8:36_4_1_1_1_2"/><i><font color="Green" title="E56">A31</font></i>: 
<a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E4:36_4_1_1_1_2"><i><font color="Green" title="E53">A28</font></i></a>, <a class="txt" href="jordan22.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E9:36_4_1_1_1_2"/><i><font color="Green" title="E57">A32</font></i>: 
<font color="Maroon" title="c11">t</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E8:36_4_1_1_1_2"><i><font color="Green" title="E56">A31</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>;<br/><a NAME="E10:36_4_1_1_1_2"/><i><font color="Green" title="E58">A33</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E18:36"><i><font color="Green" title="E46">A21</font></i></a>, <a class="txt" href="jordan22.html#E19:36"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:36_4_1_1_1_2"/>
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jordan22.html#E7:36_4_1_1_1_2"><i><font color="Green" title="E55">A30</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E12:36_4_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan22.html#E6:36"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="jordan22.html#E5:36_4_1_1_1_2"><i><font color="Green" title="E54">A29</font></i></a>, <a class="txt" href="jordan22.html#E9:36_4_1_1_1_2"><i><font color="Green" title="E57">A32</font></i></a>, <a class="txt" href="jordan22.html#E10:36_4_1_1_1_2"><i><font color="Green" title="E58">A33</font></i></a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c10">S'</font>, <font color="Maroon" title="c11">C'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:36_4_1"/><i><font color="Green" title="E52">A34</font></i>: 
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">S'</font> &amp; <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">C'</font> )
 <b>and </b><br/><a NAME="E5:36_4_1"/><i><font color="Green" title="E53">A35</font></i>: 
 <a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>,<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K9" title="WEIERSTR:func.9">min_dist_min</a> <font color="Maroon" title="c10">S'</font>,<font color="Maroon" title="c11">C'</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#D1" title="JORDAN1K:def.1">JORDAN1K:def 1</a></i>;<br/><a NAME="E6:36_4_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_4_1_2"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:36_4_1_2"/><i><font color="Green" title="E54">A36</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E4:36_4_1_2"/><i><font color="Green" title="E55">A37</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">x</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> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan22.html#E3:36_4_1_2"><i><font color="Green" title="E54">A36</font></i></a></i>;<br/>
<a NAME="E6:36_4_1_2"/><i><font color="Green" title="E56">A38</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan22.html#E4:36_4_1_2"><i><font color="Green" title="E55">A37</font></i></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E7:36_4_1_2"/><i><font color="Green" title="E57">A39</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E6:36"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="jordan22.html#E8:36"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="jordan22.html#E22:36"><i><font color="Green" title="E49">A24</font></i></a>, <a class="txt" href="jordan22.html#E3:36_4_1_2"><i><font color="Green" title="E54">A36</font></i></a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/>
<a NAME="E8:36_4_1_2"/><b>then </b><i><font color="Green" title="E58">A40</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E2:36_4_1"><i><font color="Green" title="E51">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:36_4_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="jordan22.html#E7:36"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="jordan22.html#E9:36"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="jordan22.html#E11:36"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="jordan22.html#E3:36_4_1_2"><i><font color="Green" title="E54">A36</font></i></a>, <a class="ref" href="sppol_1.html#D3" title="SPPOL_1:def.3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E10:36_4_1_2"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E3:36"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="jordan22.html#E4:36"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="jordan22.html#E10:36"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="jordan22.html#E6:36_4_1_2"><i><font color="Green" title="E56">A38</font></i></a>, <a class="txt" href="jordan22.html#E8:36_4_1_2"><i><font color="Green" title="E58">A40</font></i></a>, <a class="ref" href="goboard7.html#T8" title="GOBOARD7:th.8">GOBOARD7:8</a></i>;<br/>
<a NAME="E11:36_4_1_2"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E12:36"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_2"><i><font color="Green" title="E55">A37</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:36_4_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan22.html#E2:36_4_1"><i><font color="Green" title="E51">A26</font></i></a>, <a class="txt" href="jordan22.html#E7:36_4_1_2"><i><font color="Green" title="E57">A39</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:36_4_1"/><b>then </b>
 <a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>,<font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan22.html#E4:36_4_1"><i><font color="Green" title="E52">A34</font></i></a>, <a class="txt" href="jordan22.html#E5:36_4_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="ref" href="jgraph_1.html#T55" title="JGRAPH_1:th.55">JGRAPH_1:55</a></i>;<br/><a NAME="E8:36_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:36_4_1"/><i><font color="Green" title="E54">A41</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">k</font>
 <b>and </b><br/><a NAME="E11:36_4_1"/><i><font color="Green" title="E55">A42</font></i>: 
(  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 <b>by </b><i><a class="ref" href="gobrd14.html#T21" title="GOBRD14:th.21">GOBRD14:21</a></i>;<br/><b>set </b><font color="Maroon" title="c13">f</font> =  <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font>;<br/><b>set </b><font color="Maroon" title="c14">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font>;<br/><a NAME="E12:36_4_1"/><i><font color="Green" title="E56">A43</font></i>: 
<span class="p1">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36_4_1"><i><font color="Green" title="E55">A42</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>set </b><font color="Maroon" title="c15">s</font> =  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E13:36_4_1"/><i><font color="Green" title="E57">A44</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="jordan22.html#E13:36"><i><font color="Green" title="E43">A11</font></i></a></i>;<br/><a NAME="E14:36_4_1"/><i><font color="Green" title="E58">A45</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan21.html#T61" title="JORDAN21:th.61">JORDAN21:61</a></i>;<br/><a NAME="E15:36_4_1"/>
 <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E10:36_4_1"><i><font color="Green" title="E54">A41</font></i></a>, <a class="ref" href="jordan1a.html#T90" title="JORDAN1A:th.90">JORDAN1A:90</a></i>;<br/><a NAME="E16:36_4_1"/><b>then </b><i><font color="Green" title="E59">A46</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E14:36_4_1"><i><font color="Green" title="E58">A45</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:36_4_1"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E14:36"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="jordan22.html#E10:36_4_1"><i><font color="Green" title="E54">A41</font></i></a></i>;<br/><a NAME="E18:36_4_1"/><b>then </b>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan22.html#E17:36"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/><a NAME="E19:36_4_1"/><b>then </b><i><font color="Green" title="E60">A47</font></i>: 
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jordan22.html#E5:36"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="jordan22.html#E6:36"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="jordan22.html#E7:36"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="jordan22.html#E8:36"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="jordan22.html#E13:36_4_1"><i><font color="Green" title="E57">A44</font></i></a>, <a class="txt" href="jordan22.html#E16:36_4_1"><i><font color="Green" title="E59">A46</font></i></a>, <a class="ref" href="goboard7.html#T8" title="GOBOARD7:th.8">GOBOARD7:8</a></i>;<br/><a NAME="E20:36_4_1"/><i><font color="Green" title="E61">A48</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan6.html#T76" title="JORDAN6:th.76">JORDAN6:76</a></i>;<br/><a NAME="E21:36_4_1"/>
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan21.html#T44" title="JORDAN21:th.44">JORDAN21:44</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E23:36_4_1"/><i><font color="Green" title="E62">A49</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> )
 <b>and </b><br/><a NAME="E24:36_4_1"/><i><font color="Green" title="E63">A50</font></i>: 
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E20:36_4_1"><i><font color="Green" title="E61">A48</font></i></a>, <a class="ref" href="sppol_2.html#T13" title="SPPOL_2:th.13">SPPOL_2:13</a></i>;<br/><a NAME="E25:36_4_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="ref" href="jordan9.html#T33" title="JORDAN9:th.33">JORDAN9:33</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E27:36_4_1"/><i><font color="Green" title="E64">A51</font></i>: 
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>and </b><br/><a NAME="E28:36_4_1"/><i><font color="Green" title="E65">A52</font></i>: 
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">c</font> = <font color="Maroon" title="c17">c</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> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan22.html#E28:36_4_1"><i><font color="Green" title="E65">A52</font></i></a></i>;<br/><a NAME="E30:36_4_1"/><i><font color="Green" title="E66">A53</font></i>: 
 <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="ref" href="jordan9.html#D1" title="JORDAN9:def.1">JORDAN9:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">i1</font>, <font color="Maroon" title="c20">j1</font>, <font color="Maroon" title="c21">i2</font>, <font color="Maroon" title="c22">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E32:36_4_1"/><i><font color="Green" title="E67">A54</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>and </b><br/><a NAME="E33:36_4_1"/><i><font color="Green" title="E68">A55</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font>
 <b>and </b><br/><a NAME="E34:36_4_1"/><i><font color="Green" title="E69">A56</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>and </b><br/><a NAME="E35:36_4_1"/><i><font color="Green" title="E70">A57</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font>
 <b>and </b><br/><a NAME="E36:36_4_1"/><i><font color="Green" title="E71">A58</font></i>: 
( ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E37:36_4_1"/>
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="jordan22.html#E30:36_4_1"><i><font color="Green" title="E66">A53</font></i></a>, <a class="ref" href="gobrd13.html#T30" title="GOBRD13:th.30">GOBRD13:30</a></i>;<br/><a NAME="E38:36_4_1"/><b>then </b><i><font color="Green" title="E72">A59</font></i>: 
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E24:36_4_1"><i><font color="Green" title="E63">A50</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E39:36_4_1"/><i><font color="Green" title="E73">A60</font></i>: 
<font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E40:36_4_1"/><i><font color="Green" title="E74">A61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">i2</font>
 ;<br/><a NAME="E41:36_4_1"/><i><font color="Green" title="E75">A62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">j2</font>
 ;<br/><a NAME="E42:36_4_1"/><i><font color="Green" title="E76">A63</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E43:36_4_1"/><i><font color="Green" title="E77">A64</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E44:36_4_1"/><i><font color="Green" title="E78">A65</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i2</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E45:36_4_1"/><i><font color="Green" title="E79">A66</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">j2</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E46:36_4_1"/><i><font color="Green" title="E80">A67</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E47:36_4_1"/><i><font color="Green" title="E81">A68</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E48:36_4_1"/><i><font color="Green" title="E82">A69</font></i>: 
<font color="Maroon" title="c19">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E49:36_4_1"/><i><font color="Green" title="E83">A70</font></i>: 
<font color="Maroon" title="c20">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E50:36_4_1"/><i><font color="Green" title="E84">A71</font></i>: 
<font color="Maroon" title="c21">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E51:36_4_1"/><i><font color="Green" title="E85">A72</font></i>: 
<font color="Maroon" title="c22">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">s'</font> =  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon" title="c24">c'</font> = <font color="Maroon" title="c18">c</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/><a NAME="E53:36_4_1"/><i><font color="Green" title="E86">A73</font></i>: 
 <a href="weierstr.html#K9" title="WEIERSTR:func.9">min_dist_min</a> <font color="Maroon" title="c10">S'</font>,<font color="Maroon" title="c11">C'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c23">s'</font>,<font color="Maroon" title="c24">c'</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E4:36_4_1"><i><font color="Green" title="E52">A34</font></i></a>, <a class="txt" href="jordan22.html#E19:36_4_1"><i><font color="Green" title="E60">A47</font></i></a>, <a class="txt" href="jordan22.html#E28:36_4_1"><i><font color="Green" title="E65">A52</font></i></a>, <a class="ref" href="weierstr.html#T40" title="WEIERSTR:th.40">WEIERSTR:40</a></i>;<br/><a NAME="E54:36_4_1"/><i><font color="Green" title="E87">A74</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c23">s'</font>,<font color="Maroon" title="c24">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">c</font>
 <b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/><a NAME="E55:36_4_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T6" target="_self" title="JORDAN22:th.6">Th6</a></i>;<br/><a NAME="E56:36_4_1"/><b>then </b><i><font color="Green" title="E88">A75</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T5" target="_self" title="JORDAN22:th.5">Th5</a>, <a class="ref" href="gobrd14.html#T19" title="GOBRD14:th.19">GOBRD14:19</a></i>;<br/><a NAME="E57:36_4_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T7" target="_self" title="JORDAN22:th.7">Th7</a></i>;<br/><a NAME="E58:36_4_1"/><b>then </b><i><font color="Green" title="E89">A76</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T5" target="_self" title="JORDAN22:th.5">Th5</a>, <a class="ref" href="gobrd14.html#T20" title="GOBRD14:th.20">GOBRD14:20</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_4_1_3_1"/>
( ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> ) or ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan22.html#E36:36_4_1"><i><font color="Green" title="E71">A58</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_1_3_1_1"><b>suppose </b></a><a NAME="E1:36_4_1_3_1_1"/><i><font color="Green" title="E90">A77</font></i>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> )
 ;<br/><div class="add"><a NAME="E2:36_4_1_3_1_1"/><b>then </b><i><font color="Green" title="E91">A78</font></i>: 
<font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E3:36_4_1_3_1_1"/>
<font color="Maroon" title="c19">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E33:36_4_1"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E35:36_4_1"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_1"><i><font color="Green" title="E90">A77</font></i></a>, <a class="ref" href="jordan10.html#T1" title="JORDAN10:th.1">JORDAN10:1</a></i>;<br/><a NAME="E4:36_4_1_3_1_1"/><b>then </b><i><font color="Green" title="E92">A79</font></i>: 
<font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:36_4_1_3_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="jordan22.html#E30:36_4_1"><i><font color="Green" title="E66">A53</font></i></a>, <a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E33:36_4_1"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E35:36_4_1"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="jordan22.html#E39:36_4_1"><i><font color="Green" title="E73">A60</font></i></a>, <a class="txt" href="jordan22.html#E41:36_4_1"><i><font color="Green" title="E75">A62</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_1"><i><font color="Green" title="E90">A77</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><a NAME="E6:36_4_1_3_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c18">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c18">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan22.html#E27:36_4_1"><i><font color="Green" title="E64">A51</font></i></a>, <a class="txt" href="jordan22.html#E38:36_4_1"><i><font color="Green" title="E72">A59</font></i></a>, <a class="txt" href="jordan22.html#E42:36_4_1"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="jordan22.html#E43:36_4_1"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="jordan22.html#E2:36_4_1_3_1_1"><i><font color="Green" title="E91">A78</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_3_1_1"><i><font color="Green" title="E92">A79</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></i>;<br/><a NAME="E7:36_4_1_3_1_1"/><b>then </b><i><font color="Green" title="E93">A80</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal6.html#T104" title="TOPREAL6:th.104">TOPREAL6:104</a></i>;<br/><a NAME="E8:36_4_1_3_1_1"/><i><font color="Green" title="E94">A81</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E43:36_4_1"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="jordan22.html#E46:36_4_1"><i><font color="Green" title="E80">A67</font></i></a>, <a class="txt" href="jordan22.html#E49:36_4_1"><i><font color="Green" title="E83">A70</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_3_1_1"><i><font color="Green" title="E92">A79</font></i></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></i>;<br/><a NAME="E9:36_4_1_3_1_1"/><b>then </b><i><font color="Green" title="E95">A82</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="ref" href="gobrd14.html#T15" title="GOBRD14:th.15">GOBRD14:15</a></i>;<br/><a NAME="E10:36_4_1_3_1_1"/><i><font color="Green" title="E96">A83</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_1"><i><font color="Green" title="E90">A77</font></i></a>, <a class="ref" href="gobrd14.html#T16" title="GOBRD14:th.16">GOBRD14:16</a></i>;<br/><a NAME="E11:36_4_1_3_1_1"/><i><font color="Green" title="E97">A84</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_1"><i><font color="Green" title="E90">A77</font></i></a>, <a class="ref" href="gobrd14.html#T19" title="GOBRD14:th.19">GOBRD14:19</a></i>;<br/><a NAME="E12:36_4_1_3_1_1"/>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<font color="Maroon" title="c20">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E8:36_4_1_3_1_1"><i><font color="Green" title="E94">A81</font></i></a>, <a class="ref" href="gobrd14.html#T20" title="GOBRD14:th.20">GOBRD14:20</a></i>;<br/><b>hence </b><a NAME="E13:36_4_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan22.html#E5:36_4_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="jordan22.html#E12:36_4_1"><i><font color="Green" title="E56">A43</font></i></a>, <a class="txt" href="jordan22.html#E53:36_4_1"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="jordan22.html#E54:36_4_1"><i><font color="Green" title="E87">A74</font></i></a>, <a class="txt" href="jordan22.html#E56:36_4_1"><i><font color="Green" title="E88">A75</font></i></a>, <a class="txt" href="jordan22.html#E58:36_4_1"><i><font color="Green" title="E89">A76</font></i></a>, <a class="txt" href="jordan22.html#E7:36_4_1_3_1_1"><i><font color="Green" title="E93">A80</font></i></a>, <a class="txt" href="jordan22.html#E9:36_4_1_3_1_1"><i><font color="Green" title="E95">A82</font></i></a>, <a class="txt" href="jordan22.html#E10:36_4_1_3_1_1"><i><font color="Green" title="E96">A83</font></i></a>, <a class="txt" href="jordan22.html#E11:36_4_1_3_1_1"><i><font color="Green" title="E97">A84</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_1_3_1_2"><b>suppose </b></a><a NAME="E1:36_4_1_3_1_2"/><i><font color="Green" title="E90">A85</font></i>: 
( <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> )
 ;<br/><div class="add"><a NAME="E2:36_4_1_3_1_2"/><b>then </b><i><font color="Green" title="E91">A86</font></i>: 
<font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E3:36_4_1_3_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">j1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E33:36_4_1"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E35:36_4_1"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_2"><i><font color="Green" title="E90">A85</font></i></a>, <a class="ref" href="jordan10.html#T3" title="JORDAN10:th.3">JORDAN10:3</a></i>;<br/><a NAME="E4:36_4_1_3_1_2"/><b>then </b><i><font color="Green" title="E92">A87</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E5:36_4_1_3_1_2"/><i><font color="Green" title="E93">A88</font></i>: 
<font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E49:36_4_1"><i><font color="Green" title="E83">A70</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E6:36_4_1_3_1_2"/><i><font color="Green" title="E94">A89</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E4:36_4_1_3_1_2"><i><font color="Green" title="E92">A87</font></i></a>, <a class="ref" href="binarith.html#T61" title="BINARITH:th.61">BINARITH:61</a></i>;<br/><a NAME="E7:36_4_1_3_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c19">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="jordan22.html#E30:36_4_1"><i><font color="Green" title="E66">A53</font></i></a>, <a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E33:36_4_1"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E35:36_4_1"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="jordan22.html#E39:36_4_1"><i><font color="Green" title="E73">A60</font></i></a>, <a class="txt" href="jordan22.html#E40:36_4_1"><i><font color="Green" title="E74">A61</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_2"><i><font color="Green" title="E90">A85</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><a NAME="E8:36_4_1_3_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c18">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c18">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan22.html#E27:36_4_1"><i><font color="Green" title="E64">A51</font></i></a>, <a class="txt" href="jordan22.html#E38:36_4_1"><i><font color="Green" title="E72">A59</font></i></a>, <a class="txt" href="jordan22.html#E42:36_4_1"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="jordan22.html#E49:36_4_1"><i><font color="Green" title="E83">A70</font></i></a>, <a class="txt" href="jordan22.html#E2:36_4_1_3_1_2"><i><font color="Green" title="E91">A86</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_3_1_2"><i><font color="Green" title="E92">A87</font></i></a>, <a class="txt" href="jordan22.html#E6:36_4_1_3_1_2"><i><font color="Green" title="E94">A89</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></i>;<br/><a NAME="E9:36_4_1_3_1_2"/><b>then </b><i><font color="Green" title="E95">A90</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal6.html#T104" title="TOPREAL6:th.104">TOPREAL6:104</a></i>;<br/><a NAME="E10:36_4_1_3_1_2"/><i><font color="Green" title="E96">A91</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E42:36_4_1"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="jordan22.html#E48:36_4_1"><i><font color="Green" title="E82">A69</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_3_1_2"><i><font color="Green" title="E92">A87</font></i></a>, <a class="txt" href="jordan22.html#E5:36_4_1_3_1_2"><i><font color="Green" title="E93">A88</font></i></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></i>;<br/><a NAME="E11:36_4_1_3_1_2"/><i><font color="Green" title="E97">A92</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E44:36_4_1"><i><font color="Green" title="E78">A65</font></i></a>, <a class="txt" href="jordan22.html#E50:36_4_1"><i><font color="Green" title="E84">A71</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_2"><i><font color="Green" title="E90">A85</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_3_1_2"><i><font color="Green" title="E92">A87</font></i></a>, <a class="txt" href="jordan22.html#E5:36_4_1_3_1_2"><i><font color="Green" title="E93">A88</font></i></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></i>;<br/><a NAME="E12:36_4_1_3_1_2"/><b>then </b><i><font color="Green" title="E98">A93</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E10:36_4_1_3_1_2"><i><font color="Green" title="E96">A91</font></i></a>, <a class="ref" href="gobrd14.html#T15" title="GOBRD14:th.15">GOBRD14:15</a></i>;<br/><a NAME="E13:36_4_1_3_1_2"/><i><font color="Green" title="E99">A94</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E6:36_4_1_3_1_2"><i><font color="Green" title="E94">A89</font></i></a>, <a class="txt" href="jordan22.html#E10:36_4_1_3_1_2"><i><font color="Green" title="E96">A91</font></i></a>, <a class="ref" href="gobrd14.html#T16" title="GOBRD14:th.16">GOBRD14:16</a></i>;<br/><a NAME="E14:36_4_1_3_1_2"/><i><font color="Green" title="E100">A95</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E6:36_4_1_3_1_2"><i><font color="Green" title="E94">A89</font></i></a>, <a class="txt" href="jordan22.html#E10:36_4_1_3_1_2"><i><font color="Green" title="E96">A91</font></i></a>, <a class="ref" href="gobrd14.html#T19" title="GOBRD14:th.19">GOBRD14:19</a></i>;<br/><a NAME="E15:36_4_1_3_1_2"/>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E10:36_4_1_3_1_2"><i><font color="Green" title="E96">A91</font></i></a>, <a class="txt" href="jordan22.html#E11:36_4_1_3_1_2"><i><font color="Green" title="E97">A92</font></i></a>, <a class="ref" href="gobrd14.html#T20" title="GOBRD14:th.20">GOBRD14:20</a></i>;<br/><b>hence </b><a NAME="E16:36_4_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan22.html#E5:36_4_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="jordan22.html#E12:36_4_1"><i><font color="Green" title="E56">A43</font></i></a>, <a class="txt" href="jordan22.html#E53:36_4_1"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="jordan22.html#E54:36_4_1"><i><font color="Green" title="E87">A74</font></i></a>, <a class="txt" href="jordan22.html#E56:36_4_1"><i><font color="Green" title="E88">A75</font></i></a>, <a class="txt" href="jordan22.html#E58:36_4_1"><i><font color="Green" title="E89">A76</font></i></a>, <a class="txt" href="jordan22.html#E9:36_4_1_3_1_2"><i><font color="Green" title="E95">A90</font></i></a>, <a class="txt" href="jordan22.html#E12:36_4_1_3_1_2"><i><font color="Green" title="E98">A93</font></i></a>, <a class="txt" href="jordan22.html#E13:36_4_1_3_1_2"><i><font color="Green" title="E99">A94</font></i></a>, <a class="txt" href="jordan22.html#E14:36_4_1_3_1_2"><i><font color="Green" title="E100">A95</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_1_3_1_3"><b>suppose </b></a><a NAME="E1:36_4_1_3_1_3"/><i><font color="Green" title="E90">A96</font></i>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> )
 ;<br/><div class="add"><a NAME="E2:36_4_1_3_1_3"/><b>then </b><i><font color="Green" title="E91">A97</font></i>: 
<font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E3:36_4_1_3_1_3"/>
<font color="Maroon" title="c20">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E33:36_4_1"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E35:36_4_1"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_3"><i><font color="Green" title="E90">A96</font></i></a>, <a class="ref" href="jordan10.html#T4" title="JORDAN10:th.4">JORDAN10:4</a></i>;<br/><a NAME="E4:36_4_1_3_1_3"/><b>then </b><i><font color="Green" title="E92">A98</font></i>: 
<font color="Maroon" title="c20">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:36_4_1_3_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="jordan22.html#E30:36_4_1"><i><font color="Green" title="E66">A53</font></i></a>, <a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E33:36_4_1"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E35:36_4_1"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="jordan22.html#E39:36_4_1"><i><font color="Green" title="E73">A60</font></i></a>, <a class="txt" href="jordan22.html#E40:36_4_1"><i><font color="Green" title="E74">A61</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_3"><i><font color="Green" title="E90">A96</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><a NAME="E6:36_4_1_3_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c18">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c18">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan22.html#E27:36_4_1"><i><font color="Green" title="E64">A51</font></i></a>, <a class="txt" href="jordan22.html#E38:36_4_1"><i><font color="Green" title="E72">A59</font></i></a>, <a class="txt" href="jordan22.html#E43:36_4_1"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="jordan22.html#E44:36_4_1"><i><font color="Green" title="E78">A65</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_3"><i><font color="Green" title="E90">A96</font></i></a>, <a class="txt" href="jordan22.html#E2:36_4_1_3_1_3"><i><font color="Green" title="E91">A97</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_3_1_3"><i><font color="Green" title="E92">A98</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></i>;<br/><a NAME="E7:36_4_1_3_1_3"/><b>then </b><i><font color="Green" title="E93">A99</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal6.html#T104" title="TOPREAL6:th.104">TOPREAL6:104</a></i>;<br/><a NAME="E8:36_4_1_3_1_3"/><i><font color="Green" title="E94">A100</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_3"><i><font color="Green" title="E90">A96</font></i></a>, <a class="ref" href="gobrd14.html#T15" title="GOBRD14:th.15">GOBRD14:15</a></i>;<br/><a NAME="E9:36_4_1_3_1_3"/><i><font color="Green" title="E95">A101</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E44:36_4_1"><i><font color="Green" title="E78">A65</font></i></a>, <a class="txt" href="jordan22.html#E47:36_4_1"><i><font color="Green" title="E81">A68</font></i></a>, <a class="txt" href="jordan22.html#E50:36_4_1"><i><font color="Green" title="E84">A71</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_3"><i><font color="Green" title="E90">A96</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_3_1_3"><i><font color="Green" title="E92">A98</font></i></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></i>;<br/><a NAME="E10:36_4_1_3_1_3"/><b>then </b><i><font color="Green" title="E96">A102</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="ref" href="gobrd14.html#T16" title="GOBRD14:th.16">GOBRD14:16</a></i>;<br/><a NAME="E11:36_4_1_3_1_3"/><i><font color="Green" title="E97">A103</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E9:36_4_1_3_1_3"><i><font color="Green" title="E95">A101</font></i></a>, <a class="ref" href="gobrd14.html#T19" title="GOBRD14:th.19">GOBRD14:19</a></i>;<br/><a NAME="E12:36_4_1_3_1_3"/>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i2</font>,<font color="Maroon" title="c22">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_3"><i><font color="Green" title="E90">A96</font></i></a>, <a class="ref" href="gobrd14.html#T20" title="GOBRD14:th.20">GOBRD14:20</a></i>;<br/><b>hence </b><a NAME="E13:36_4_1_3_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan22.html#E5:36_4_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="jordan22.html#E12:36_4_1"><i><font color="Green" title="E56">A43</font></i></a>, <a class="txt" href="jordan22.html#E53:36_4_1"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="jordan22.html#E54:36_4_1"><i><font color="Green" title="E87">A74</font></i></a>, <a class="txt" href="jordan22.html#E56:36_4_1"><i><font color="Green" title="E88">A75</font></i></a>, <a class="txt" href="jordan22.html#E58:36_4_1"><i><font color="Green" title="E89">A76</font></i></a>, <a class="txt" href="jordan22.html#E7:36_4_1_3_1_3"><i><font color="Green" title="E93">A99</font></i></a>, <a class="txt" href="jordan22.html#E8:36_4_1_3_1_3"><i><font color="Green" title="E94">A100</font></i></a>, <a class="txt" href="jordan22.html#E10:36_4_1_3_1_3"><i><font color="Green" title="E96">A102</font></i></a>, <a class="txt" href="jordan22.html#E11:36_4_1_3_1_3"><i><font color="Green" title="E97">A103</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_1_3_1_4"><b>suppose </b></a><a NAME="E1:36_4_1_3_1_4"/><i><font color="Green" title="E90">A104</font></i>: 
( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i2</font> &amp; <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:36_4_1_3_1_4"/><b>then </b><i><font color="Green" title="E91">A105</font></i>: 
<font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E3:36_4_1_3_1_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">i1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E33:36_4_1"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E35:36_4_1"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_4"><i><font color="Green" title="E90">A104</font></i></a>, <a class="ref" href="jordan10.html#T2" title="JORDAN10:th.2">JORDAN10:2</a></i>;<br/><a NAME="E4:36_4_1_3_1_4"/><b>then </b><i><font color="Green" title="E92">A106</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E5:36_4_1_3_1_4"/><i><font color="Green" title="E93">A107</font></i>: 
<font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E48:36_4_1"><i><font color="Green" title="E82">A69</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E6:36_4_1_3_1_4"/><i><font color="Green" title="E94">A108</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E4:36_4_1_3_1_4"><i><font color="Green" title="E92">A106</font></i></a>, <a class="ref" href="binarith.html#T61" title="BINARITH:th.61">BINARITH:61</a></i>;<br/><a NAME="E7:36_4_1_3_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c16">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E23:36_4_1"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="jordan22.html#E30:36_4_1"><i><font color="Green" title="E66">A53</font></i></a>, <a class="txt" href="jordan22.html#E32:36_4_1"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="jordan22.html#E33:36_4_1"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="jordan22.html#E34:36_4_1"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="jordan22.html#E35:36_4_1"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="jordan22.html#E39:36_4_1"><i><font color="Green" title="E73">A60</font></i></a>, <a class="txt" href="jordan22.html#E41:36_4_1"><i><font color="Green" title="E75">A62</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_4"><i><font color="Green" title="E90">A104</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><a NAME="E8:36_4_1_3_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c18">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c18">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan22.html#E27:36_4_1"><i><font color="Green" title="E64">A51</font></i></a>, <a class="txt" href="jordan22.html#E38:36_4_1"><i><font color="Green" title="E72">A59</font></i></a>, <a class="txt" href="jordan22.html#E45:36_4_1"><i><font color="Green" title="E79">A66</font></i></a>, <a class="txt" href="jordan22.html#E48:36_4_1"><i><font color="Green" title="E82">A69</font></i></a>, <a class="txt" href="jordan22.html#E2:36_4_1_3_1_4"><i><font color="Green" title="E91">A105</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_3_1_4"><i><font color="Green" title="E92">A106</font></i></a>, <a class="txt" href="jordan22.html#E6:36_4_1_3_1_4"><i><font color="Green" title="E94">A108</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></i>;<br/><a NAME="E9:36_4_1_3_1_4"/><b>then </b><i><font color="Green" title="E95">A109</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal6.html#T104" title="TOPREAL6:th.104">TOPREAL6:104</a></i>;<br/><a NAME="E10:36_4_1_3_1_4"/><i><font color="Green" title="E96">A110</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E45:36_4_1"><i><font color="Green" title="E79">A66</font></i></a>, <a class="txt" href="jordan22.html#E51:36_4_1"><i><font color="Green" title="E85">A72</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_3_1_4"><i><font color="Green" title="E92">A106</font></i></a>, <a class="txt" href="jordan22.html#E5:36_4_1_3_1_4"><i><font color="Green" title="E93">A107</font></i></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></i>;<br/><a NAME="E11:36_4_1_3_1_4"/><i><font color="Green" title="E97">A111</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E42:36_4_1"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="jordan22.html#E45:36_4_1"><i><font color="Green" title="E79">A66</font></i></a>, <a class="txt" href="jordan22.html#E48:36_4_1"><i><font color="Green" title="E82">A69</font></i></a>, <a class="txt" href="jordan22.html#E51:36_4_1"><i><font color="Green" title="E85">A72</font></i></a>, <a class="txt" href="jordan22.html#E6:36_4_1_3_1_4"><i><font color="Green" title="E94">A108</font></i></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></i>;<br/><a NAME="E12:36_4_1_3_1_4"/><b>then </b><i><font color="Green" title="E98">A112</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E10:36_4_1_3_1_4"><i><font color="Green" title="E96">A110</font></i></a>, <a class="ref" href="gobrd14.html#T15" title="GOBRD14:th.15">GOBRD14:15</a></i>;<br/><a NAME="E13:36_4_1_3_1_4"/><i><font color="Green" title="E99">A113</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E43:36_4_1"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="jordan22.html#E49:36_4_1"><i><font color="Green" title="E83">A70</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1_3_1_4"><i><font color="Green" title="E90">A104</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_1_3_1_4"><i><font color="Green" title="E92">A106</font></i></a>, <a class="txt" href="jordan22.html#E5:36_4_1_3_1_4"><i><font color="Green" title="E93">A107</font></i></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></i>;<br/><a NAME="E14:36_4_1_3_1_4"/><b>then </b><i><font color="Green" title="E100">A114</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E10:36_4_1_3_1_4"><i><font color="Green" title="E96">A110</font></i></a>, <a class="ref" href="gobrd14.html#T16" title="GOBRD14:th.16">GOBRD14:16</a></i>;<br/><a NAME="E15:36_4_1_3_1_4"/><i><font color="Green" title="E101">A115</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E10:36_4_1_3_1_4"><i><font color="Green" title="E96">A110</font></i></a>, <a class="txt" href="jordan22.html#E13:36_4_1_3_1_4"><i><font color="Green" title="E99">A113</font></i></a>, <a class="ref" href="gobrd14.html#T19" title="GOBRD14:th.19">GOBRD14:19</a></i>;<br/><a NAME="E16:36_4_1_3_1_4"/>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E10:36_4_1_3_1_4"><i><font color="Green" title="E96">A110</font></i></a>, <a class="txt" href="jordan22.html#E11:36_4_1_3_1_4"><i><font color="Green" title="E97">A111</font></i></a>, <a class="ref" href="gobrd14.html#T20" title="GOBRD14:th.20">GOBRD14:20</a></i>;<br/><b>hence </b><a NAME="E17:36_4_1_3_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan22.html#E5:36_4_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="jordan22.html#E12:36_4_1"><i><font color="Green" title="E56">A43</font></i></a>, <a class="txt" href="jordan22.html#E53:36_4_1"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="jordan22.html#E54:36_4_1"><i><font color="Green" title="E87">A74</font></i></a>, <a class="txt" href="jordan22.html#E56:36_4_1"><i><font color="Green" title="E88">A75</font></i></a>, <a class="txt" href="jordan22.html#E58:36_4_1"><i><font color="Green" title="E89">A76</font></i></a>, <a class="txt" href="jordan22.html#E9:36_4_1_3_1_4"><i><font color="Green" title="E95">A109</font></i></a>, <a class="txt" href="jordan22.html#E12:36_4_1_3_1_4"><i><font color="Green" title="E98">A112</font></i></a>, <a class="txt" href="jordan22.html#E14:36_4_1_3_1_4"><i><font color="Green" title="E100">A114</font></i></a>, <a class="txt" href="jordan22.html#E15:36_4_1_3_1_4"><i><font color="Green" title="E101">A115</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E60:36_4_1"/>
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_2"><b>suppose </b></a><a NAME="E1:36_4_2"/>
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:36_4_2"/><b>then </b><i><font color="Green" title="E50">A116</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E3:36_4_2"/><i><font color="Green" title="E51">A117</font></i>: 
 <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan19.html#D4" title="JORDAN19:def.4">JORDAN19:def 4</a></i>;<br/><a NAME="E4:36_4_2"/>
 ex <font color="Olive" title="b1">S</font> being  <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of 2 st <br/>( <font color="Olive" title="b1">S</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a> &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b2">x</font> ) &amp;  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Olive" title="b1">S</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c10">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p4">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> </span>}</span> ;<br/>
<a NAME="E1:36_4_2_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p4">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_4_2_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p4">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> </span>}</span> 
 ;<br/>

<a NAME="E2:36_4_2_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> )
 
;<br/>
<b>hence </b><a NAME="E3:36_4_2_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p4">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E3:36_4_2_1"/><i><font color="Green" title="E52">A118</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 
;<br/>
<a NAME="E4:36_4_2_1"/><i><font color="Green" title="E53">A119</font></i>: 
<font color="Maroon" title="c11">R</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_1_2"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a><br/>

<b>let </b><font color="Maroon" title="c12">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:36_4_2_1_2"/>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 ;<br/>

<a NAME="E2:36_4_2_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> )
 
;<br/>
<a NAME="E3:36_4_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E14:36"><i><font color="Green" title="E44">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:36_4_2_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">r</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E17:36"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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> 2</span>)</span> =  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,$1</span>)</span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c12">S</font> being   <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of 2<b> such that </b><br/><a NAME="E6:36_4_2_1"/><i><font color="Green" title="E54">A120</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="toprns_1.html#S1" title="TOPRNS_1:sch.1">TOPRNS_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c12">S</font>
;<br/>

<a NAME="E7:36_4_2_1"/><i><font color="Green" title="E55">A121</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:36_4_2_1_3"/><i><font color="Green" title="E56">A122</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">r</font>
 ;<br/>

<a NAME="E2:36_4_2_1_3"/><i><font color="Green" title="E57">A123</font></i>: 
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> holds <br/> <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:36_4_2_1_3_1"/>
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 ;<br/>

<a NAME="E2:36_4_2_1_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> )
 
;<br/>
<a NAME="E3:36_4_2_1_3_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E14:36"><i><font color="Green" title="E44">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:36_4_2_1_3_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">r</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E17:36"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36_4_2_1_3"/>
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:36_4_2_1_3_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:36_4_2_1_3_2"/><i><font color="Green" title="E58">A124</font></i>: 
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span>
 <b>and </b><br/><a NAME="E4:36_4_2_1_3_2"/><i><font color="Green" title="E59">A125</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E17:36"><i><font color="Green" title="E45">A20</font></i></a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">r</font>
;<br/>

<b>consider </b><font color="Maroon" title="c16">x</font> being   <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> 2</span>)</span><b> such that </b><br/><a NAME="E6:36_4_2_1_3_2"/><i><font color="Green" title="E60">A126</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font>
 <b>and </b><br/><a NAME="E7:36_4_2_1_3_2"/><i><font color="Green" title="E61">A127</font></i>: 
<a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E3:36_4_2_1_3_2"><i><font color="Green" title="E58">A124</font></i></a>, <a class="txt" href="jordan22.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E8:36_4_2_1_3_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U</font>
 
<b>by </b><i><a class="txt" href="jordan22.html#E6:36_4_2_1_3_2"><i><font color="Green" title="E60">A126</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:36_4_2_1_3_2"/><i><font color="Green" title="E62">A128</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c17">n</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E11:36_4_2_1_3_2"/><i><font color="Green" title="E63">A129</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">n</font>
 ;<br/>
<a NAME="E12:36_4_2_1_3_2"/>
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c17">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E7:36_4_2_1_3_2"><i><font color="Green" title="E61">A127</font></i></a>, <a class="txt" href="jordan22.html#E10:36_4_2_1_3_2"><i><font color="Green" title="E62">A128</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>;<br/>
<b>hence </b><a NAME="E13:36_4_2_1_3_2"/>
( <font color="Maroon" title="c15">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">r</font> )
 <b>by </b><i><a class="txt" href="jordan22.html#E4:36_4_2_1_3_2"><i><font color="Green" title="E59">A125</font></i></a>, <a class="txt" href="jordan22.html#E11:36_4_2_1_3_2"><i><font color="Green" title="E63">A129</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:36_4_2_1_3"/><b>then </b><i><font color="Green" title="E58">A130</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c11">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">U</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E3:36_4_2_1"><i><font color="Green" title="E52">A118</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_2_1"><i><font color="Green" title="E53">A119</font></i></a>, <a class="txt" href="jordan22.html#E2:36_4_2_1_3"><i><font color="Green" title="E57">A123</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">v</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E6:36_4_2_1_3"/><i><font color="Green" title="E59">A131</font></i>: 
<font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 <b>and </b><br/><a NAME="E7:36_4_2_1_3"/><i><font color="Green" title="E60">A132</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c11">R</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">v</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E3:36_4_2_1"><i><font color="Green" title="E52">A118</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_2_1"><i><font color="Green" title="E53">A119</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_2_1_3"><i><font color="Green" title="E56">A122</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E8:36_4_2_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c11">R</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">v</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c13">r</font>
 
<b>by </b><i><a class="txt" href="jordan22.html#E7:36_4_2_1_3"><i><font color="Green" title="E60">A132</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:36_4_2_1_3"/><b>then </b><i><font color="Green" title="E61">A133</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c11">R</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">v</font>
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:36_4_2_1_3"/><i><font color="Green" title="E62">A134</font></i>: 
<font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>and </b><br/><a NAME="E12:36_4_2_1_3"/><i><font color="Green" title="E63">A135</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">n</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E6:36_4_2_1_3"><i><font color="Green" title="E59">A131</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">n</font>
;<br/>

<b>let </b><font color="Maroon" title="c16">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E13:36_4_2_1_3"/><i><font color="Green" title="E64">A136</font></i>: 
<font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c16">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E6:36_4_2_1"><i><font color="Green" title="E54">A120</font></i></a></i>;<br/>
<b>assume </b><a NAME="E14:36_4_2_1_3"/><i><font color="Green" title="E65">A137</font></i>: 
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">m</font>
 ;<br/>

<a NAME="E15:36_4_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c16">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E12:36_4_2_1_3"><i><font color="Green" title="E63">A135</font></i></a>, <a class="ref" href="jordan22.html#T30" target="_self" title="JORDAN22:th.30">Th30</a></i>;<br/>
<a NAME="E16:36_4_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">v</font>
 
<b>by </b><i><a class="txt" href="jordan22.html#E6:36_4_2_1"><i><font color="Green" title="E54">A120</font></i></a>, <a class="txt" href="jordan22.html#E11:36_4_2_1_3"><i><font color="Green" title="E62">A134</font></i></a></i>;<br/>
<a NAME="E17:36_4_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">v</font>
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E18:36_4_2_1_3"/><b>then </b><i><font color="Green" title="E66">A138</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">r</font>
 
<b>by </b><i><a class="txt" href="jordan22.html#E2:36_4_2"><i><font color="Green" title="E50">A116</font></i></a>, <a class="txt" href="jordan22.html#E4:36_4_2_1_3"><i><font color="Green" title="E58">A130</font></i></a>, <a class="txt" href="jordan22.html#E9:36_4_2_1_3"><i><font color="Green" title="E61">A133</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:36_4_2_1_3"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c16">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E12:36_4_2_1_3"><i><font color="Green" title="E63">A135</font></i></a>, <a class="txt" href="jordan22.html#E14:36_4_2_1_3"><i><font color="Green" title="E65">A137</font></i></a>, <a class="ref" href="jordan22.html#T26" target="_self" title="JORDAN22:th.26">Th26</a></i>;<br/>
<a NAME="E20:36_4_2_1_3"/><b>then </b><i><font color="Green" title="E67">A139</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E13:36_4_2_1_3"><i><font color="Green" title="E64">A136</font></i></a>, <a class="ref" href="real_2.html#T105" title="REAL_2:th.105">REAL_2:105</a></i>;<br/>
<a NAME="E21:36_4_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="jordan22.html#E13:36"><i><font color="Green" title="E43">A11</font></i></a>, <a class="txt" href="jordan22.html#E13:36_4_2_1_3"><i><font color="Green" title="E64">A136</font></i></a></i>;<br/>
<a NAME="E22:36_4_2_1_3"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E10:36"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E23:36_4_2_1_3"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T49" title="JGRAPH_1:th.49">JGRAPH_1:49</a></i>;<br/>
<a NAME="E24:36_4_2_1_3"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E20:36_4_2_1_3"><i><font color="Green" title="E67">A139</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E25:36_4_2_1_3"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">r</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E18:36_4_2_1_3"><i><font color="Green" title="E66">A138</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E8:36_4_2_1"/><i><font color="Green" title="E56">A140</font></i>: 
<font color="Maroon" title="c12">S</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_1_4"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="toprns_1.html#D9" title="TOPRNS_1:def.9">TOPRNS_1:def 9</a><br/>

<b>thus </b><a NAME="E1:36_4_2_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  ex <font color="Olive">b<sub>2</sub></font> being   <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>  st <br/>for <font color="Olive">b<sub>3</sub></font> being   <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>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan22.html#E7:36_4_2_1"><i><font color="Green" title="E55">A121</font></i></a></i>;<br/>


</div><b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:36_4_2_1_5"/><i><font color="Green" title="E57">A141</font></i>: 
<font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E6:36_4_2_1"><i><font color="Green" title="E54">A120</font></i></a></i>;<br/><a NAME="E2:36_4_2_1_5"/>
<span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan19.html#D2" title="JORDAN19:def.2">JORDAN19:def 2</a></i>;<br/><b>hence </b><a NAME="E3:36_4_2_1_5"/>
<font color="Maroon" title="c12">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E1:36_4_2_1_5"><i><font color="Green" title="E57">A141</font></i></a>, <a class="ref" href="jordan21.html#T44" title="JORDAN21:th.44">JORDAN21:44</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E10:36_4_2_1"/>
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c12">S</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E7:36_4_2_1"><i><font color="Green" title="E55">A121</font></i></a>, <a class="txt" href="jordan22.html#E8:36_4_2_1"><i><font color="Green" title="E56">A140</font></i></a>, <a class="ref" href="toprns_1.html#D10" title="TOPRNS_1:def.10">TOPRNS_1:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:36_4_2"/>
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E3:36_4_2"><i><font color="Green" title="E51">A117</font></i></a>, <a class="ref" href="kurato_2.html#T58" title="KURATO_2:th.58">KURATO_2:58</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
