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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c3">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="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:89"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font>) <b>-&gt; </b>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> = <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> $1</span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">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="complsp1.html#K8" title="COMPLSP1:func.8">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>  <b>from </b><i><a class="ref" href="complsp1.html#S1" target="_self" title="COMPLSP1:sch.1">COMPLSP1:sch 1</a>();<br/></i>
<a NAME="E3:89"/><i><font color="Green" title="E56">A2</font></i>: 
<span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="complsp1.html#E1:89"><i><font color="Green" title="E55">A1</font></i></a></i>;<br/>
<div><i><font color="Green" title="E57">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:89_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/><a NAME="E2:89_1"/><b>then </b>
 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="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</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/><b>hence </b><a NAME="E3:89_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T65" target="_self" title="COMPLSP1:th.65">Th65</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:89"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon" title="c4">X</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>proof </b></a><div class="add">

<b>take </b>
0
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a><br/>

<b>thus </b><a NAME="E1:89_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="complsp1.html#E4:89"><i><font color="Green" title="E57">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E59">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">r1</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:89_3"/><i><font color="Green" title="E60">A6</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r1</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">r</font> = <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><b>take </b><font color="Maroon" title="c7">r</font> = <font color="Maroon" title="c6">r</font>;<br/><b>thus </b><a NAME="E3:89_3"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="complsp1.html#E1:89"><i><font color="Green" title="E55">A1</font></i></a></i>;<br/><b>thus </b><a NAME="E4:89_3"/>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">r1</font>
 <b>by </b><i><a class="txt" href="complsp1.html#E1:89_3"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="complsp1.html#T72" target="_self" title="COMPLSP1:th.72">Th72</a></i>;<br/></div><b>end;</b></div>
<b>thus </b> <a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">A</font> = 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="ref" href="complsp1.html#D18" target="_self" title="COMPLSP1:def.18">Def18</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="complsp1.html#E3:89"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="complsp1.html#E4:89"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="complsp1.html#E5:89"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="complsp1.html#E6:89"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>
;<br/>


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