<?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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</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="b2">A</font>, <font color="Olive" title="b3">B</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="b2">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>  &amp; <font color="Olive" title="b3">B</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/><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="b2">A</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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">B</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="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>  &amp; <font color="Olive" title="b2">B</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/><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">(<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="b2">B</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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>  &amp; <font color="Maroon" title="c4">B</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 <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="c3">A</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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">B</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> )</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">B</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"><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/>
<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">z</font>) where <font color="Olive" title="b1">z</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">z</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="seq_4.html#S2" target="_self" title="SEQ_4:sch.2">SEQ_4:sch 2</a>();<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <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="c3">A</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</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="c3">A</font>;<br/>
<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>,   <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">$1 <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> $2</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="c6">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</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>3</sub></font>[<font color="Olive" title="b1">z</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="seq_4.html#S2" target="_self" title="SEQ_4:sch.2">SEQ_4:sch 2</a>();<br/></span>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:171"/><span class="lab"><font color="Green" title="E84">A1</font></span>: 
<font color="Maroon" title="c3">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> 
 <span class="kw">and </span><br/><a NAME="E4:171"/><span class="lab"><font color="Green" title="E85">A2</font></span>: 
<font color="Maroon" title="c4">B</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="Red">::  thesis: </font></span></a><span class="hide"> <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="c3">A</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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">B</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><br/>

<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="E6:171"/><span class="lab"><font color="Green" title="E86">A3</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">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:171"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/>
<a NAME="E7:171"/><span class="lab"><font color="Green" title="E87">A4</font></span>: 
( <font color="Maroon" title="c5">Y</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>  &amp; <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="171_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:171_1"/>
<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="c5">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:171"><span class="lab"><font color="Green" title="E86">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:171_1"/>
<font color="Maroon" title="c5">Y</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">Y</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> </span><br/>

<span class="kw">take </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seq_4.html#D2" target="_self" title="SEQ_4:def.2">SEQ_4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> implies  <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="c8">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:171_1"/>
<font color="Maroon" title="c8">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="Red">::  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="c8">r</font></span><br/>

<a NAME="E4:171_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="c8">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"><font color="Maroon" title="c2">x</font> <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">B</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:171_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="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T112" target="_self" title="SEQ_4:th.112">Th65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:171"/><span class="lab"><font color="Green" title="E88">A5</font></span>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">A</font> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D20" target="_self" title="SEQ_4:def.20">Def18</a></span>;<br/>
<a NAME="E9:171"/><span class="lab"><font color="Green" title="E89">A6</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="c5">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:171"><span class="lab"><font color="Green" title="E86">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">Z</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z</font>) where <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z</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>2</sub></font>[<font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z</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="seq_4.html#S3" target="_self" title="SEQ_4:sch.3">SEQ_4:sch 3</a>();<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c9">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><span class="kw"> such that </span><br/><a NAME="E12:171"/><span class="lab"><font color="Green" title="E90">A7</font></span>: 
<font color="Maroon" title="c9">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:171"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E91">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="171_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">r</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</a> <font color="Maroon" title="c5">Y</font> implies  ex <font color="Olive" title="b1">r3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:171_2"/>
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</a> <font color="Maroon" title="c5">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">r2</font>, <font color="Maroon" title="c12">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:171_2"/><span class="lab"><font color="Green" title="E92">A9</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">r1</font>
 <span class="kw">and </span><br/><a NAME="E4:171_2"/><span class="lab"><font color="Green" title="E93">A10</font></span>: 
<font color="Maroon" title="c11">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">and </span><br/><a NAME="E5:171_2"/><span class="lab"><font color="Green" title="E94">A11</font></span>: 
<font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c13">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="E7:171_2"/><span class="lab"><font color="Green" title="E95">A12</font></span>: 
( <font color="Maroon" title="c12">r1</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="c13">z2</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Maroon" title="c13">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:171_2"><span class="lab"><font color="Green" title="E94">A11</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">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><span class="kw"> such that </span><br/><a NAME="E9:171_2"/><span class="lab"><font color="Green" title="E96">A13</font></span>: 
<font color="Maroon" title="c11">r2</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="c14">z1</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="E10:171_2"/><span class="lab"><font color="Green" title="E97">A14</font></span>: 
<font color="Maroon" title="c14">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:171_2"><span class="lab"><font color="Green" title="E93">A10</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c15">r3</font> = <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="c14">z1</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c13">z2</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c15">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font> )</span><br/><a NAME="E11:171_2"/>
<font color="Maroon" title="c11">r2</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="c14">z1</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c2">x</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="txt" href="#E9:171_2"><span class="lab"><font color="Green" title="E96">A13</font></span></a>, <a class="ref" href="seq_4.html#T121" target="_self" title="SEQ_4:th.121">Th74</a></span>;<br/><span class="kw">hence </span><a NAME="E12:171_2"/>
( <font color="Maroon" title="c15">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c15">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:171_2"><span class="lab"><font color="Green" title="E92">A9</font></span></a>, <a class="txt" href="#E10:171_2"><span class="lab"><font color="Green" title="E97">A14</font></span></a>, <a class="txt" href="#E7:171_2"><span class="lab"><font color="Green" title="E95">A12</font></span></a>, <a class="ref" href="seq_4.html#T122" target="_self" title="SEQ_4:th.122">Th75</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:171"/><span class="lab"><font color="Green" title="E92">A15</font></span>: 
<font color="Maroon" title="c8">Z</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="171_3"><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="Red">:: according to </font></span><a class="ref" href="seq_4.html#D2" target="_self" title="SEQ_4:def.2">SEQ_4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:171_3"/>
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c10">r</font></span><br/>

<a NAME="E2:171_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">z1</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> st <br/>( <font color="Maroon" title="c10">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"><font color="Olive" title="b1">z1</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Olive" title="b2">z</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="c3">A</font> &amp; <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:171_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T112" target="_self" title="SEQ_4:th.112">Th65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:171"/><span class="lab"><font color="Green" title="E93">A16</font></span>: 
( <font color="Maroon" title="c6">X</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>  &amp; <font color="Maroon" title="c6">X</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="171_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:171_4"/>
<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">z1</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="#E12:171"><span class="lab"><font color="Green" title="E90">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:171_4"/>
<font color="Maroon" title="c6">X</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> </span><br/>

<span class="kw">take </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seq_4.html#D2" target="_self" title="SEQ_4:def.2">SEQ_4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:171_4"/>
<font color="Maroon" title="c10">r</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="Red">::  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="c10">r</font></span><br/>

<a NAME="E4:171_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c10">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"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:171_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T112" target="_self" title="SEQ_4:th.112">Th65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:171"/><span class="lab"><font color="Green" title="E94">A17</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D22" target="_self" title="SEQ_4:def.22">Def20</a></span>;<br/>
<a NAME="E17:171"/>
<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">z1</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="#E12:171"><span class="lab"><font color="Green" title="E90">A7</font></span></a></span>;<br/>
<a NAME="E18:171"/><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"><font color="Maroon" title="c2">x</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c9">z1</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></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"><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> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</a> <font color="Maroon" title="c5">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:171"><span class="lab"><font color="Green" title="E89">A6</font></span></a></span>;<br/>
<a NAME="E19:171"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c8">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:171"><span class="lab"><font color="Green" title="E92">A15</font></span></a>, <a class="txt" href="#E13:171"><span class="lab"><font color="Green" title="E91">A8</font></span></a>, <a class="ref" href="seq_4.html#T143" target="_self" title="SEQ_4:th.143">Th97</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:171"/>
<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="c3">A</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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">B</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:171"><span class="lab"><font color="Green" title="E93">A16</font></span></a>, <a class="txt" href="#E7:171"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E8:171"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="txt" href="#E16:171"><span class="lab"><font color="Green" title="E94">A17</font></span></a>, <a class="ref" href="seq_4.html#T142" target="_self" title="SEQ_4:th.142">Th96</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
