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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font><br/>  for <font color="Olive" title="b3">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Olive" title="b2">z</font></span>)</span>,<font color="Olive" title="b3">A</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">A</font>)</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Olive" title="b2">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">z</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Olive" title="b1">A</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">A</font>)</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Maroon" title="c4">A</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> )</span><br/>







<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>) <span class="kw">-&gt; </span>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> = <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> $1</span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">Y</font> = <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z1</font>) where <font color="Olive" title="b1">z1</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z1</font>] </span> } </span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S8" title="DOMAIN_1:sch.8">DOMAIN_1:sch 8</a>();<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>) <span class="kw">-&gt; </span>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> = <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> $1</span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span>;<br/>
<a NAME="E2:134"/><span class="lab"><font color="Green" title="E73">A1</font></span>: 
<font color="Maroon" title="c5">Y</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D9" title="XXREAL_2:def.9">XXREAL_2:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c5">Y</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">r</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:134_1"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font></span><br/>

<a NAME="E2:134_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Olive" title="b1">z1</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:134_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T95" target="_self" title="SEQ_4:th.95">Th112</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c6">X</font> = <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">z1</font>) where <font color="Olive" title="b1">z1</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z1</font>] </span> } </span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S8" title="DOMAIN_1:sch.8">DOMAIN_1:sch 8</a>();<br/></span>
<span class="kw">assume </span><a NAME="E4:134"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Maroon" title="c4">A</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">z2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E6:134"/><span class="lab"><font color="Green" title="E74">A2</font></span>: 
<font color="Maroon" title="c7">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T4" title="SUBSET_1:th.4">SUBSET_1:4</a></span>;<br/>
<a NAME="E7:134"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
 <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Maroon" title="c4">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c5">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D17" target="_self" title="SEQ_4:def.17">Def20</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E76">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">r9</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">r9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> implies <font color="Maroon" title="c8">r9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:134_2"/>
<font color="Maroon" title="c8">r9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">r9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">z3</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:134_2"/><span class="lab"><font color="Green" title="E77">A5</font></span>: 
<font color="Maroon" title="c8">r9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c9">z3</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span>
 <span class="kw">and </span><br/><a NAME="E4:134_2"/><span class="lab"><font color="Green" title="E78">A6</font></span>: 
<font color="Maroon" title="c9">z3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><a NAME="E5:134_2"/>
<span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c9">z3</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c9">z3</font></span>)</span> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T76" target="_self" title="SEQ_4:th.76">Th93</a></span>;<br/><a NAME="E6:134_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A7</font></span>: 
<span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c9">z3</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r9</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:134_2"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="ref" href="seq_4.html#T98" target="_self" title="SEQ_4:th.98">Th115</a></span>;<br/><a NAME="E7:134_2"/>
<span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c9">z3</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:134_2"><span class="lab"><font color="Green" title="E78">A6</font></span></a></span>;<br/><a NAME="E8:134_2"/><span class="kw">then </span>
<span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c9">z3</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Maroon" title="c4">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:134"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="txt" href="#E2:134"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="seq_4.html#D2" target="_self" title="SEQ_4:def.2">Def5</a></span>;<br/><a NAME="E9:134_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">r9</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Maroon" title="c4">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:134_2"><span class="lab"><font color="Green" title="E79">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E10:134_2"/>
<font color="Maroon" title="c8">r9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:134"/><span class="lab"><font color="Green" title="E77">A8</font></span>: 
<span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c7">z2</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:134"><span class="lab"><font color="Green" title="E74">A2</font></span></a></span>;<br/>
<a NAME="E10:134"/>
 <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D17" target="_self" title="SEQ_4:def.17">Def20</a></span>;<br/>
<a NAME="E11:134"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:134"><span class="lab"><font color="Green" title="E77">A8</font></span></a>, <a class="txt" href="#E8:134"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="ref" href="seq_4.html#T113" target="_self" title="SEQ_4:th.113">Th130</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:134"/>
 <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z</font></span>)</span>,<font color="Maroon" title="c4">A</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">A</font>)</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
