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

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/>

<b>set </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<b>set </b><font color="Maroon">p</font> =  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">U</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">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">LMP</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> </span>}</span> 
 ;<br/>

<a NAME="E2:36_1"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> )
 
;<br/>
<b>hence </b><a NAME="E3:36_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">U</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon">n0</font> =  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">q</font> =  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E3:36"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E5:36"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E6:36"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E8:36"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E9:36"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E10:36"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E11:36"/><i><font color="Green">E68</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i>, <a class="txt" href="jordan22.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E12:36"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="jordan21.html#T48">JORDAN21:48</a></i>;<br/>
<a NAME="E13:36"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:36_2"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1g.html#T41">JORDAN1G:41</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="ref" href="jordan21.html#T28">JORDAN21:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="txt" href="jordan22.html#E4:36"><i><font color="Green">E49</font></i></a>, <a class="ref" href="jordan21.html#T29">JORDAN21:29</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E14:36"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">s</font> =  <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:36_3"/><i><font color="Green">E73</font></i>: 
 <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan21.html#T61">JORDAN21:61</a></i>;<br/>
<b>assume </b><a NAME="E2:36_3"/><i><font color="Green">E74</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:36_3"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:36_3"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:36_3"/><b>then </b>
 <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1a.html#T90">JORDAN1A:90</a></i>;<br/>
<a NAME="E6:36_3"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan22.html#T2" target="_self">Th2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:36_3"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="jordan22.html#E3:36"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E8:36_3"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:36_3"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, , , , , <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/>
<a NAME="E10:36_3"/>
 <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:36_3"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:36_3"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">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">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">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">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T98">TOPREAL6:98</a></i>;<br/>
<a NAME="E16:36"/><b>then </b>
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="jct_misc.html#T23">JCT_MISC:23</a></i>;<br/>
<a NAME="E17:36"/><b>then </b><i><font color="Green">E96</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E18:36"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan21.html#T61">JORDAN21:61</a></i>;<br/>
<a NAME="E19:36"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E20:36"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E4:36"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E21:36"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan22.html#T2" target="_self">Th2</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E22:36"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">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">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> or  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</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">E101</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><div><i><font color="Green">E102</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">E123</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">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">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan22.html#E8"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="real_1.html#D5">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">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">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="ref" href="euclid.html#T56">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">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:36_4_1_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:36_4_1_1_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span>
 <b>and </b><br/><a NAME="E5:36_4_1_1_1_2"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan22.html#T2" target="_self">Th2</a>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:36_4_1_1_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font>
 <b>and </b><br/><a NAME="E8:36_4_1_1_1_2"/><i><font color="Green">E127</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan22.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E9:36_4_1_1_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan22.html#T8" target="_self">Th8</a>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/><a NAME="E10:36_4_1_1_1_2"/><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:36_4_1_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jordan22.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D3">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="ref" href="jordan22.html#T6" target="_self">Th6</a>, <a class="txt" href="jordan22.html#E5:36"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan22.html#E6:36"><i><font color="Green">E63</font></i></a>, <a class="ref" href="topreal1.html#T10">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">S'</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span></span>)</span>, <font color="Maroon">C'</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:36_4_1"/><i><font color="Green">E131</font></i>: 
(  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">S'</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C'</font> )
 <b>and </b><br/><a NAME="E5:36_4_1"/><i><font color="Green">E132</font></i>: 
 <a href="jordan1k.html#K4">dist_min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>,<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">S'</font>,<font color="Maroon">C'</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#D1">JORDAN1K:def 1</a></i>;<br/><a NAME="E6:36_4_1"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">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">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R2">meets</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_4_1_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E4:36_4_1_2"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan22.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E6:36_4_1_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan22.html#T7" target="_self">Th7</a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E7:36_4_1_2"/><i><font color="Green">E146</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="txt" href="jordan22.html#E2"><i><font color="Green">Lemma43</font></i></a>, , <a class="ref" href="jordan22.html#T6" target="_self">Th6</a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<a NAME="E8:36_4_1_2"/><b>then </b><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E7"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:36_4_1_2"/>
<font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="jordan22.html#E1"><i><font color="Green">Lemma42</font></i></a>, , , <a class="ref" href="jordan22.html#T6" target="_self">Th6</a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E10:36_4_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, , , <a class="ref" href="jordan22.html#T8" target="_self">Th8</a>, <a class="txt" href="jordan22.html#E6:36"><i><font color="Green">E63</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/>
<a NAME="E11:36_4_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="jordan22.html#T1" target="_self">Th1</a>, <a class="ref" href="jordan22.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D3">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#E7"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="jordan22.html#E5:36"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D1">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">dist_min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>,<font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan22.html#E8"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="jordan22.html#T5" target="_self">Th5</a>, , <a class="ref" href="jgraph_1.html#T55">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">dist_min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:36_4_1"/><i><font color="Green">E148</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E11:36_4_1"/><i><font color="Green">E149</font></i>: 
(  <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp;  <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> 2,1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i><a class="ref" href="gobrd14.html#T21">GOBRD14:21</a></i>;<br/><b>set </b><font color="Maroon">f</font> =  <a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font>;<br/><b>set </b><font color="Maroon">G</font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font>;<br/><a NAME="E12:36_4_1"/><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>set </b><font color="Maroon">s</font> =  <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E13:36_4_1"/><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="jordan22.html#E3:36"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E14:36_4_1"/><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan21.html#T61">JORDAN21:61</a></i>;<br/><a NAME="E15:36_4_1"/>
 <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan1a.html#T90">JORDAN1A:90</a></i>;<br/><a NAME="E16:36_4_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E6:36"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:36_4_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="jordan22.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E18:36_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E4:36"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E19:36_4_1"/><b>then </b><i><font color="Green">E154</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan22.html#T2" target="_self">Th2</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><a NAME="E20:36_4_1"/><i><font color="Green">E155</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, , <a class="txt" href="jordan22.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="jordan22.html#E2"><i><font color="Green">Lemma43</font></i></a>, <a class="txt" href="jordan22.html#E5:36"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan22.html#E7:36"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan22.html#E8:36"><i><font color="Green">E65</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/><a NAME="E21:36_4_1"/><i><font color="Green">E156</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan6.html#T76">JORDAN6:76</a></i>;<br/><a NAME="E22:36_4_1"/>
 <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E24:36_4_1"/><i><font color="Green">E157</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> )
 <b>and </b><br/><a NAME="E25:36_4_1"/><i><font color="Green">E158</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E10:36"><i><font color="Green">E67</font></i></a>, <a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/><a NAME="E26:36_4_1"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan9.html#T33">JORDAN9:33</a></i>;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E28:36_4_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E29:36_4_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/><a NAME="E31:36_4_1"/><i><font color="Green">E161</font></i>: 
 <a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="jordan9.html#D1">JORDAN9:def 1</a></i>;<br/><b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E33:36_4_1"/><i><font color="Green">E166</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E34:36_4_1"/><i><font color="Green">E167</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E35:36_4_1"/><i><font color="Green">E168</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E36:36_4_1"/><i><font color="Green">E169</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>and </b><br/><a NAME="E37:36_4_1"/><i><font color="Green">E170</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan22.html#E14:36"><i><font color="Green">E72</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E38:36_4_1"/>
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan22.html#E14:36"><i><font color="Green">E72</font></i></a>, <a class="ref" href="gobrd13.html#T30">GOBRD13:30</a></i>;<br/><a NAME="E39:36_4_1"/><b>then </b><i><font color="Green">E171</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E12:36"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E40:36_4_1"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 0
 ;<br/><a NAME="E41:36_4_1"/><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font>
 ;<br/><a NAME="E42:36_4_1"/><i><font color="Green">E174</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j2</font>
 ;<br/><a NAME="E43:36_4_1"/><i><font color="Green">E175</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E44:36_4_1"/><i><font color="Green">E176</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font>
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E45:36_4_1"/><i><font color="Green">E177</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E46:36_4_1"/><i><font color="Green">E178</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E47:36_4_1"/><i><font color="Green">E179</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E48:36_4_1"/><i><font color="Green">E180</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E49:36_4_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E50:36_4_1"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E51:36_4_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E52:36_4_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>reconsider </b><font color="Maroon">s'</font> =  <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon">c'</font> = <font color="Maroon">c</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/><a NAME="E54:36_4_1"/><i><font color="Green">E187</font></i>: 
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">S'</font>,<font color="Maroon">C'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">s'</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E8"><i><font color="Green">Lemma57</font></i></a>, <a class="txt" href="jordan22.html#E9:36"><i><font color="Green">E66</font></i></a>, , , <a class="ref" href="weierstr.html#T40">WEIERSTR:40</a></i>;<br/><a NAME="E55:36_4_1"/><i><font color="Green">E188</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">s'</font>,<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a>  <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/><a NAME="E56:36_4_1"/><i><font color="Green">E189</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,1</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E57:36_4_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E58:36_4_1"/><b>then </b><i><font color="Green">E190</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="gobrd14.html#T19">GOBRD14:19</a></i>;<br/><a NAME="E59:36_4_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E60:36_4_1"/><b>then </b><i><font color="Green">E191</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="gobrd14.html#T20">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">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><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">E192</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:36_4_1_3_1_1"/><b>then </b><i><font color="Green">E193</font></i>: 
<font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E3:36_4_1_3_1_1"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, , , <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan22.html#T10" target="_self">Th10</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jordan10.html#T1">JORDAN10:1</a></i>;<br/><a NAME="E4:36_4_1_3_1_1"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:36_4_1_3_1_1"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan22.html#E14:36"><i><font color="Green">E72</font></i></a>, , , <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan22.html#T10" target="_self">Th10</a>, , , <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="gobrd13.html#D2">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan22.html#E13:36"><i><font color="Green">E70</font></i></a>, <a class="ref" href="jordan22.html#T11" target="_self">Th11</a>, , , <a class="txt" href="jordan22.html#E19:36"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jordan9.html#T19">JORDAN9:19</a></i>;<br/><a NAME="E7:36_4_1_3_1_1"/><b>then </b><i><font color="Green">E195</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p4">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gobrd14.html#T12">GOBRD14:12</a></i>;<br/><a NAME="E8:36_4_1_3_1_1"/><i><font color="Green">E196</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan22.html#T14" target="_self">Th14</a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E9:36_4_1_3_1_1"/><b>then </b><i><font color="Green">E197</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="gobrd14.html#T15">GOBRD14:15</a></i>;<br/><a NAME="E10:36_4_1_3_1_1"/><i><font color="Green">E198</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="gobrd14.html#T16">GOBRD14:16</a></i>;<br/><a NAME="E11:36_4_1_3_1_1"/><i><font color="Green">E199</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="gobrd14.html#T19">GOBRD14:19</a></i>;<br/><a NAME="E12:36_4_1_3_1_1"/>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan22.html#E1:36_4_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="gobrd14.html#T20">GOBRD14:20</a></i>;<br/><b>hence </b><a NAME="E13:36_4_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan22.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan22.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan22.html#T17" target="_self">Th17</a>, , , <a class="txt" href="jordan22.html#E17:36"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan22.html#E22:36"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan22.html#E2:36_4_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jordan22.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan22.html#T19" target="_self">Th19</a>, <a class="ref" href="xxreal_0.html#T2">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">E200</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:36_4_1_3_1_2"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E3:36_4_1_3_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, , , <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan22.html#T10" target="_self">Th10</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jordan10.html#T3">JORDAN10:3</a></i>;<br/><a NAME="E4:36_4_1_3_1_2"/><b>then </b><i><font color="Green">E202</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E5:36_4_1_3_1_2"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E6:36_4_1_3_1_2"/><i><font color="Green">E204</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jordan3.html#T6">JORDAN3:6</a></i>;<br/><a NAME="E7:36_4_1_3_1_2"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan22.html#E14:36"><i><font color="Green">E72</font></i></a>, , , <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan22.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan22.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="gobrd13.html#D2">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan22.html#E13:36"><i><font color="Green">E70</font></i></a>, <a class="ref" href="jordan22.html#T11" target="_self">Th11</a>, , <a class="ref" href="jordan22.html#T14" target="_self">Th14</a>, <a class="txt" href="jordan22.html#E19:36"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jordan9.html#T19">JORDAN9:19</a></i>;<br/><a NAME="E9:36_4_1_3_1_2"/><b>then </b><i><font color="Green">E205</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p4">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p4">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gobrd14.html#T12">GOBRD14:12</a></i>;<br/><a NAME="E10:36_4_1_3_1_2"/><i><font color="Green">E206</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jordan22.html#E22:36"><i><font color="Green">E100</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E11:36_4_1_3_1_2"/><i><font color="Green">E207</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan22.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jordan22.html#E22:36"><i><font color="Green">E100</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E12:36_4_1_3_1_2"/><b>then </b><i><font color="Green">E208</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T18" target="_self">Th18</a>, <a class="ref" href="gobrd14.html#T15">GOBRD14:15</a></i>;<br/><a NAME="E13:36_4_1_3_1_2"/><i><font color="Green">E209</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan22.html#E1:36_4_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jordan22.html#T18" target="_self">Th18</a>, <a class="ref" href="gobrd14.html#T16">GOBRD14:16</a></i>;<br/><a NAME="E14:36_4_1_3_1_2"/><i><font color="Green">E210</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan22.html#E1:36_4_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jordan22.html#T18" target="_self">Th18</a>, <a class="ref" href="gobrd14.html#T19">GOBRD14:19</a></i>;<br/><a NAME="E15:36_4_1_3_1_2"/>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan22.html#T19" target="_self">Th19</a>, <a class="ref" href="gobrd14.html#T20">GOBRD14:20</a></i>;<br/><b>hence </b><a NAME="E16:36_4_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan22.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan22.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan22.html#T17" target="_self">Th17</a>, , , <a class="txt" href="jordan22.html#E17:36"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan22.html#E2:36_4_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jordan22.html#T20" target="_self">Th20</a>, <a class="ref" href="jordan22.html#T21" target="_self">Th21</a>, <a class="ref" href="jordan22.html#T22" target="_self">Th22</a>, <a class="ref" href="xxreal_0.html#T2">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">E211</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:36_4_1_3_1_3"/><b>then </b><i><font color="Green">E212</font></i>: 
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E3:36_4_1_3_1_3"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, , , <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan22.html#T10" target="_self">Th10</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jordan10.html#T4">JORDAN10:4</a></i>;<br/><a NAME="E4:36_4_1_3_1_3"/><b>then </b><i><font color="Green">E213</font></i>: 
<font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:36_4_1_3_1_3"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan22.html#E14:36"><i><font color="Green">E72</font></i></a>, , , <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan22.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan22.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="gobrd13.html#D2">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan22.html#E13:36"><i><font color="Green">E70</font></i></a>, <a class="ref" href="jordan22.html#T11" target="_self">Th11</a>, , , <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan22.html#E19:36"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jordan9.html#T19">JORDAN9:19</a></i>;<br/><a NAME="E7:36_4_1_3_1_3"/><b>then </b><i><font color="Green">E214</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p4">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gobrd14.html#T12">GOBRD14:12</a></i>;<br/><a NAME="E8:36_4_1_3_1_3"/><i><font color="Green">E215</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="gobrd14.html#T15">GOBRD14:15</a></i>;<br/><a NAME="E9:36_4_1_3_1_3"/><i><font color="Green">E216</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan22.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E10:36_4_1_3_1_3"/><b>then </b><i><font color="Green">E217</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="gobrd14.html#T16">GOBRD14:16</a></i>;<br/><a NAME="E11:36_4_1_3_1_3"/><i><font color="Green">E218</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="txt" href="jordan22.html#E2:36_4_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="gobrd14.html#T19">GOBRD14:19</a></i>;<br/><a NAME="E12:36_4_1_3_1_3"/>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="gobrd14.html#T20">GOBRD14:20</a></i>;<br/><b>hence </b><a NAME="E13:36_4_1_3_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan22.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan22.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan22.html#T17" target="_self">Th17</a>, , , <a class="txt" href="jordan22.html#E17:36"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan22.html#E22:36"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jordan22.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan22.html#T19" target="_self">Th19</a>, <a class="ref" href="xxreal_0.html#T2">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">E219</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:36_4_1_3_1_4"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E3:36_4_1_3_1_4"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, , , <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan22.html#T10" target="_self">Th10</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jordan10.html#T2">JORDAN10:2</a></i>;<br/><a NAME="E4:36_4_1_3_1_4"/><b>then </b><i><font color="Green">E221</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E5:36_4_1_3_1_4"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E6:36_4_1_3_1_4"/><i><font color="Green">E223</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jordan3.html#T6">JORDAN3:6</a></i>;<br/><a NAME="E7:36_4_1_3_1_4"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan22.html#E14:36"><i><font color="Green">E72</font></i></a>, , , <a class="ref" href="jordan22.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan22.html#T10" target="_self">Th10</a>, , , <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="ref" href="gobrd13.html#D2">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan22.html#E13:36"><i><font color="Green">E70</font></i></a>, <a class="ref" href="jordan22.html#T11" target="_self">Th11</a>, , , <a class="txt" href="jordan22.html#E19:36"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jordan22.html#E1:36_4_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jordan9.html#T19">JORDAN9:19</a></i>;<br/><a NAME="E9:36_4_1_3_1_4"/><b>then </b><i><font color="Green">E224</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gobrd14.html#T12">GOBRD14:12</a></i>;<br/><a NAME="E10:36_4_1_3_1_4"/><i><font color="Green">E225</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan22.html#T16" target="_self">Th16</a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jordan22.html#E22:36"><i><font color="Green">E100</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E11:36_4_1_3_1_4"/><i><font color="Green">E226</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="jordan22.html#T16" target="_self">Th16</a>, <a class="txt" href="jordan22.html#E1:36_4_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E12:36_4_1_3_1_4"/><b>then </b><i><font color="Green">E227</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T18" target="_self">Th18</a>, <a class="ref" href="gobrd14.html#T15">GOBRD14:15</a></i>;<br/><a NAME="E13:36_4_1_3_1_4"/><i><font color="Green">E228</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan22.html#T14" target="_self">Th14</a>, <a class="txt" href="jordan22.html#E18:36"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan22.html#E20:36"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jordan22.html#E22:36"><i><font color="Green">E100</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E14:36_4_1_3_1_4"/><b>then </b><i><font color="Green">E229</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T18" target="_self">Th18</a>, <a class="ref" href="gobrd14.html#T16">GOBRD14:16</a></i>;<br/><a NAME="E15:36_4_1_3_1_4"/><i><font color="Green">E230</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan22.html#T21" target="_self">Th21</a>, <a class="ref" href="gobrd14.html#T19">GOBRD14:19</a></i>;<br/><a NAME="E16:36_4_1_3_1_4"/>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan22.html#T19" target="_self">Th19</a>, <a class="ref" href="gobrd14.html#T20">GOBRD14:20</a></i>;<br/><b>hence </b><a NAME="E17:36_4_1_3_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan22.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan22.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan22.html#T17" target="_self">Th17</a>, , , <a class="txt" href="jordan22.html#E17:36"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan22.html#E2:36_4_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jordan22.html#T20" target="_self">Th20</a>, <a class="ref" href="jordan22.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan22.html#T23" target="_self">Th23</a>, <a class="ref" href="xxreal_0.html#T2">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="E62:36_4_1"/>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan19.html#K4">South_Arc</a> <font color="Maroon">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">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:36_4_2"/><b>then </b><i><font color="Green">E231</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:36_4_2"/><i><font color="Green">E232</font></i>: 
 <a href="jordan19.html#K4">South_Arc</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K8">Lim_inf</a> <span class="p1">(<span class="default"><a href="jordan19.html#K2">Lower_Appr</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan19.html#D4">JORDAN19:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">e</font> =  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/><a NAME="E5:36_4_2"/>
 ex <font color="Olive">S</font> being  <a href="toprns_1.html#NM1">Real_Sequence</a> of 2 st <br/>( <font color="Olive">S</font> is <a href="toprns_1.html#V3">convergent</a> &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan19.html#K2">Lower_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Olive">x</font> ) &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="toprns_1.html#K7">lim</a> <font color="Olive">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">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">LMP</a> <span class="p4">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">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">LMP</a> <span class="p4">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_4_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">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">LMP</a> <span class="p4">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> </span>}</span> 
 ;<br/>

<a NAME="E2:36_4_2_1_1"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> )
 
;<br/>
<b>hence </b><a NAME="E3:36_4_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">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">LMP</a> <span class="p4">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E3:36_4_2_1"/><i><font color="Green">E234</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
;<br/>
<a NAME="E4:36_4_2_1"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">R</font> is <a href="seq_4.html#V1">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">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a><br/>

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

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

<a NAME="E2:36_4_2_1_2"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> )
 
;<br/>
<a NAME="E3:36_4_2_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E4:36"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:36_4_2_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan22.html#T2" target="_self">Th2</a>, <a class="ref" href="seq_4.html#D4">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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> =  <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">S</font> being   <a href="toprns_1.html#NM1">Real_Sequence</a> of 2<b> such that </b><br/><a NAME="E6:36_4_2_1"/><i><font color="Green">E236</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="toprns_1.html#S1">TOPRNS_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<a NAME="E7:36_4_2_1"/><i><font color="Green">E237</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">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">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:36_4_2_1_3"/><i><font color="Green">E238</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:36_4_2_1_3"/><i><font color="Green">E239</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/> <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">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">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

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

<a NAME="E2:36_4_2_1_3_1"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> )
 
;<br/>
<a NAME="E3:36_4_2_1_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E4:36"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:36_4_2_1_3_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan22.html#T2" target="_self">Th2</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36_4_2_1_3"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">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">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:36_4_2_1_3_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:36_4_2_1_3_2"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span>
 <b>and </b><br/><a NAME="E4:36_4_2_1_3_2"/><i><font color="Green">E241</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan22.html#T2" target="_self">Th2</a>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:36_4_2_1_3_2"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font>
 <b>and </b><br/><a NAME="E7:36_4_2_1_3_2"/><i><font color="Green">E243</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E6:36"><i><font color="Green">E63</font></i></a>, </i>;<br/>
<a NAME="E8:36_4_2_1_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 
<b>by </b><i><a class="txt" href="jordan22.html#E8:36"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:36_4_2_1_3_2"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">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">E245</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/>
<a NAME="E12:36_4_2_1_3_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E9:36"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan22.html#E10:36"><i><font color="Green">E67</font></i></a>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<b>hence </b><a NAME="E13:36_4_2_1_3_2"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="txt" href="jordan22.html#E7:36"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</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">E246</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p4"><a href="euclid.html#K23">|[</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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E8"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="jordan22.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan22.html#E5:36"><i><font color="Green">E62</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">v</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E6:36_4_2_1_3"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E7:36_4_2_1_3"/><i><font color="Green">E249</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E8"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="jordan22.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan22.html#T8" target="_self">Th8</a>, <a class="ref" href="seq_4.html#D4">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">sup</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">v</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="jordan22.html#E8:36"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:36_4_2_1_3"/><b>then </b><i><font color="Green">E250</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">R</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">v</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:36_4_2_1_3"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>and </b><br/><a NAME="E12:36_4_2_1_3"/><i><font color="Green">E252</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E7:36"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E13:36_4_2_1_3"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan22.html#T6" target="_self">Th6</a></i>;<br/>
<b>assume </b><a NAME="E14:36_4_2_1_3"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">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">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, </i>;<br/>
<a NAME="E16:36_4_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="jordan22.html#T6" target="_self">Th6</a>, <a class="txt" href="jordan22.html#E10:36"><i><font color="Green">E67</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">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E18:36_4_2_1_3"/><b>then </b><i><font color="Green">E255</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="txt" href="jordan22.html#E6:36"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan22.html#E9:36"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:36_4_2_1_3"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="jordan22.html#E11:36"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan22.html#E13:36"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E20:36_4_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E21:36_4_2_1_3"/><b>then </b><i><font color="Green">E256</font></i>: 
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E12:36"><i><font color="Green">E69</font></i></a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<a NAME="E22:36_4_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="jordan22.html#E3:36"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan22.html#E12:36"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E23:36_4_2_1_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E24:36_4_2_1_3"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T49">JGRAPH_1:49</a></i>;<br/>
<a NAME="E25:36_4_2_1_3"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan22.html#E14:36"><i><font color="Green">E72</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E26:36_4_2_1_3"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">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">E257</font></i>: 
<font color="Maroon">S</font> is <a href="toprns_1.html#V3">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">LMP</a> <font color="Maroon">C</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="toprns_1.html#D9">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">Element</a> of  <a href="numbers.html#K1">REAL</a>   holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or  ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&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">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K2">K314</a>(<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<font color="Maroon">S</font>,<font color="Olive">b<sub>3</sub></font>) <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan22.html#T7" target="_self">Th7</a></i>;<br/>


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

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:36_4_2_1_5"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K2">LMP</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan22.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E2:36_4_2_1_5"/>
<span class="p1">(<span class="default"><a href="jordan19.html#K2">Lower_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan19.html#D2">JORDAN19:def 2</a></i>;<br/><b>hence </b><a NAME="E3:36_4_2_1_5"/>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan19.html#K2">Lower_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E5:36"><i><font color="Green">E62</font></i></a>, </i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E10:36_4_2_1"/>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="toprns_1.html#K7">lim</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="jordan22.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan22.html#T8" target="_self">Th8</a>, <a class="ref" href="toprns_1.html#D10">TOPRNS_1:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:36_4_2"/>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan19.html#K4">South_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan22.html#E7"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="kurato_2.html#T58">KURATO_2:58</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
