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<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:88"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">z2</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:88"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> = <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> = <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">z1</font>) where <font color="Olive">z1</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z1</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>from </b><i><a class="ref" href="complsp1.html#S1" target="_self">COMPLSP1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">z1</font>) where <font color="Olive">z1</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z1</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>from </b><i><a class="ref" href="complsp1.html#S1" target="_self">COMPLSP1:sch 1</a>();<br/></i>
<a NAME="E6:88"/><i><font color="Green">E60</font></i>: 
 <a href="complsp1.html#K18" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="complsp1.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E7:88"/><i><font color="Green">E102</font></i>: 
 <a href="complsp1.html#K18" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="complsp1.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E8:88"/><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="complsp1.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E9:88"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">Y</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><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">SEQ_4:def 2</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="E1:88_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:88_1"/><b>then </b>
 ex <font color="Olive">z1</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font> st <br/>( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span> <a href="complsp1.html#K13" target="_self">-</a> <font color="Olive">z1</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> &amp; <font color="Olive">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
;<br/>
<b>hence </b><a NAME="E3:88_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </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="E1:88_2"/>
<font color="Maroon">r'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">z3</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:88_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">r'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z3</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> &amp; <font color="Maroon">z3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><a NAME="E4:88_2"/>
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z3</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T42" target="_self">Th42</a></i>;<br/><a NAME="E5:88_2"/><b>then </b><i><font color="Green">E111</font></i>: 
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z3</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="complsp1.html#K18" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span>,<font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T35" target="_self">Th35</a>, <a class="ref" href="complsp1.html#T41" target="_self">Th41</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E6:88_2"/>
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z3</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z3</font></span>)</span> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span>
 <b>by </b><i><a class="ref" href="complsp1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E7:88_2"/><b>then </b>
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z3</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span>
 <b>by </b><i><a class="ref" href="complsp1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E8:88_2"/><b>then </b>
<font color="Maroon">r'</font> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="complsp1.html#K18" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span>,<font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T43" target="_self">Th43</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E9:88_2"/>
<font color="Maroon">r'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" target="_self">dist</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:88"/><b>then </b>
<span class="p1">(<span class="default"><a href="complsp1.html#K18" target="_self">dist</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complsp1.html#K18" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="complsp1.html#E54"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="complsp1.html#T37" target="_self">Th37</a>, <a class="txt" href="complsp1.html#E49"><i><font color="Green">Lemma67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:88"/>
 <a href="complsp1.html#K18" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span>,<font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">A</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>


</div>
