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

<b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>assume </b><a NAME="E1:93"/><i><font color="Green">E33</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 ;<br/>

<a NAME="E2:93"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon">np</font> =  <a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="complfld.html#K3">.|</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">z</font>) where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z</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="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>set </b><font color="Maroon">q</font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font>;<br/>
<a NAME="E4:93"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
;<br/>
<a NAME="E5:93"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">D</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_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">b</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="E1:93_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E3:93_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 ;<br/>
<b>thus </b><a NAME="E4:93_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">g1</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Olive">g1</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> );<br/>
<a NAME="E6:93"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">g</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><b>proof </b></a><div class="add">

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

<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E2:93_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E3:93_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#T5" target="_self">Th5</a>, <a class="ref" href="polynom5.html#T6" target="_self">Th6</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E5:93_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">g1</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>take </b>
<font color="Maroon">g1</font>
;<br/>

<b>thus </b><a NAME="E7:93_2"/>
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/>

<b>thus </b><a NAME="E8:93_2"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">g1</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E8:93"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">G</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="cfcont_1.html#S1">CFCONT_1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E10:93"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>
 <b>and </b><br/><a NAME="E11:93"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E12:93"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:93"/><i><font color="Green">E60</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E14:93"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E15:93"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="algseq_1.html#T25">ALGSEQ_1:25</a></i>;<br/>
<a NAME="E16:93"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U2">Zero</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>;<br/>
<a NAME="E17:93"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="polynom5.html#D1" target="_self">Def1</a>, <a class="ref" href="complfld.html#T96">COMPLFLD:96</a></i>;<br/>
<a NAME="E18:93"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="real_1.html#K3">+</a> 1;<br/>

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

<b>consider </b><font color="Maroon">Gn</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E2:93_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">Gn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:93_3"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">Gn</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E4:93_3"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">Gn</font></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:93_3"/><i><font color="Green">E67</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="txt" href="polynom5.html#E4"><i><font color="Green">Lemma52</font></i></a></i>;<br/>
<a NAME="E6:93_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i>, <a class="txt" href="polynom5.html#E11:93"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E7:93_3"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="complfld.html#T97">COMPLFLD:97</a>, <a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E8:93_3"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:93_3"/><b>then </b><i><font color="Green">E131</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="real_2.html#T164">REAL_2:164</a></i>;<br/>
<a NAME="E10:93_3"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:93_3_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:93_3_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">Gn</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="polynom5.html#E4:93_3"><i><font color="Green">E66</font></i></a>, , , <a class="ref" href="polynom5.html#T8" target="_self">Th8</a>, <a class="ref" href="polynom5.html#T9" target="_self">Th9</a>, <a class="ref" href="polynom5.html#T10" target="_self">Th10</a>, <a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="txt" href="polynom5.html#E2:93_3"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E3:93_3_1"/>
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:93_3_1"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
;<br/>
<a NAME="E5:93_3_1"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="polynom5.html#T6" target="_self">Th6</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E6:93_3_1"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom5.html#T15" target="_self">Th15</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E7:93_3_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:93_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E12:93_3"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:93"/><b>then </b>
<font color="Maroon">G</font> is <a href="comseq_2.html#V1">bounded</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#D3">COMSEQ_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">G1</font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E21:93"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">G1</font> is    <a href="comseq_3.html#M1">subsequence</a> of <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E22:93"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">G1</font> is <a href="comseq_2.html#V2">convergent</a>
 <b>by </b><i><a class="ref" href="comseq_3.html#T51">COMSEQ_3:51</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z0</font> =  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">G1</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">z0</font>
;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">G1n</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <font color="Olive">G1n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Olive">G1n</font> );<br/>
<a NAME="E24:93"/><i><font color="Green">E135</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Olive">g</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_4"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">G1n</font> = <font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">G1n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>take </b>
<font color="Maroon">G1n</font>
;<br/>

<b>thus </b><a NAME="E3:93_4"/>
<font color="Maroon">G1n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/>

<b>thus </b><a NAME="E4:93_4"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">G1n</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">H</font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E26:93"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Maroon">H</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="cfcont_1.html#S1">CFCONT_1:sch 1</a>(<a class="ref" href="polynom5.html#T9" target="_self">Th9</a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E28:93"/><i><font color="Green">E137</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">Rcons</font> = <a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<a NAME="E30:93"/><i><font color="Green">E138</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Rcons</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E31:93"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">H'</font> being <a href="comseq_2.html#V2">convergent</a> <a href="comseq_1.html#NM1">Complex_Sequence</a> holds  <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Olive">H'</font></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V4">convergent</a>
 
;<br/>
<b>consider </b><font color="Maroon">Nseq</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E33:93"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">G1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Nseq</font>
 <b>by </b><i>, <a class="ref" href="comseq_3.html#D9">COMSEQ_3:def 9</a></i>;<br/>
<a NAME="E34:93"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
;<br/>
<a NAME="E35:93"/><b>then </b><i><font color="Green">E143</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="polynom5.html#T6" target="_self">Th6</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E37:93"/><i><font color="Green">E144</font></i>: 
for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">g</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font>
 <b>by </b><i>, <a class="ref" href="comseq_2.html#D4">COMSEQ_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">eg</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">g1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:93_5"/><i><font color="Green">E145</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 ;<br/><b>consider </b><font color="Maroon">fPF</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E3:93_5"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">fPF</font> <a href="funct_2.html#R1">=</a>  <a href="polynom5.html#K9" target="_self">Polynomial-Function</a> <a href="complfld.html#K1">F_Complex</a> ,<span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>
 <b>and </b><br/><a NAME="E4:93_5"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">fPF</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i><a class="txt" href="polynom5.html#E17:93"><i><font color="Green">E63</font></i></a></i>;<br/><b>consider </b><font color="Maroon">p1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:93_5"/><i><font color="Green">E148</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p1</font>
 <b>and </b><br/><a NAME="E7:93_5"/><i><font color="Green">E149</font></i>: 
for <font color="Olive">x1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a>  &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="complex1.html#K11">-</a> <font color="Maroon">g</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p1</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fPF</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">fPF</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="polynom5.html#T21" target="_self">Th21</a>, <a class="ref" href="cfcont_1.html#T61">CFCONT_1:61</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:93_5"/><i><font color="Green">E150</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">g</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p1</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T16" target="_self">Th16</a>, <a class="ref" href="polynom5.html#T23" target="_self">Th23</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E10:93_5"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E11:93_5"/><b>then </b><i><font color="Green">E151</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">g</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p1</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">G1m</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E13:93_5"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">G1m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E14:93_5"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">H</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">G1m</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E15:93_5"/><i><font color="Green">E155</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fPF</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><i><font color="Green">E156</font></i>: <font color="Maroon">H</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font> = 
<font color="Maroon">fPF</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T20" target="_self">Th20</a>, , <a class="ref" href="polynom5.html#D3" target="_self">Def3</a>, <a class="txt" href="polynom5.html#E16:93"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<font color="Maroon">fPF</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><font color="Maroon">eg</font> = 
<font color="Maroon">fPF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="polynom5.html#T20" target="_self">Th20</a>, <a class="txt" href="polynom5.html#E16:93"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<font color="Maroon">fPF</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">g</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E18:93_5"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">eg</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E27"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="polynom5.html#T24" target="_self">Th24</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E41:93"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">H</font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#D4">COMSEQ_2:def 4</a></i>;<br/>
<a NAME="E42:93"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E43:93"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">R</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="seq_4.html#T43">SEQ_4:43</a></i>;<br/>
<a NAME="E44:93"/><b>then </b><i><font color="Green">E160</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">R</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="polynom5.html#T20" target="_self">Th20</a>, <a class="ref" href="seq_2.html#T25">SEQ_2:25</a></i>;<br/>
<a NAME="E45:93"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">Rcons</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<a NAME="E46:93"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">Rcons</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="ref" href="polynom5.html#T15" target="_self">Th15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:93_6"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">Rcons</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#T15" target="_self">Th15</a></i>;<br/><b>consider </b><font color="Maroon">G1n</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E3:93_6"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">G1n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:93_6"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">H</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">G1n</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E5:93_6"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><i><font color="Green">E166</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">R</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">G1n</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a>, <a class="ref" href="polynom5.html#D3" target="_self">Def3</a></i>
;<br/><b>consider </b><font color="Maroon">gNn</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E8:93_6"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">gNn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E9:93_6"/><i><font color="Green">E168</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">gNn</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:93_6"/>
<font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T33">SEQM_3:33</a></i>;<br/><a NAME="E11:93_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:93_6"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T152">REAL_2:152</a></i>;<br/><a NAME="E13:93_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:93_6"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">gNn</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:93_6"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">gNn</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E16:93_6"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">gNn</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E17:93_6"/>
<font color="Maroon">Rcons</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">R</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="polynom5.html#T24" target="_self">Th24</a>, , , , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E48:93"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">R</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">Rcons</font>
 
<b>by </b><i><a class="ref" href="polynom5.html#T23" target="_self">Th23</a>, <a class="txt" href="polynom5.html#E27"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E49:93"/><b>then </b><i><font color="Green">E169</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">R</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/>
<i><font color="Green">E170</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">R</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">H</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T20" target="_self">Th20</a>, <a class="ref" href="polynom5.html#T21" target="_self">Th21</a>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">H</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i>, <a class="ref" href="comseq_2.html#T11">COMSEQ_2:11</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">H</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K5">-</a> 0
<b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="seq_4.html#T44">SEQ_4:44</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">enp0</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">PF</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E53:93"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">PF</font> <a href="funct_2.html#R1">=</a>  <a href="polynom5.html#K9" target="_self">Polynomial-Function</a> <a href="complfld.html#K1">F_Complex</a> ,<span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>
 <b>and </b><br/><a NAME="E54:93"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">PF</font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i><a class="txt" href="polynom5.html#E17:93"><i><font color="Green">E63</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:93_8"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:93_8"/><i><font color="Green">E173</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:93_8"/><i><font color="Green">E174</font></i>: 
for <font color="Olive">x1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a>  &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">PF</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">PF</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="cfcont_1.html#T61">CFCONT_1:61</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:93_8"/><i><font color="Green">E175</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E7:93_8"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E8:93_8"/><b>then </b><i><font color="Green">E176</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T25" target="_self">Th25</a></i>;<br/><b>consider </b><font color="Maroon">G1m</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E10:93_8"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">G1m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E11:93_8"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">H</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">G1m</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E12:93_8"/><i><font color="Green">E242</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">PF</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><i><font color="Green">E243</font></i>: <font color="Maroon">PF</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> = 
<font color="Maroon">PF</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="polynom5.html#D3" target="_self">Def3</a>, <a class="ref" href="polynom5.html#T27" target="_self">Th27</a>, <a class="ref" href="polynom5.html#T28" target="_self">Th28</a>, <a class="txt" href="polynom5.html#E16:93"><i><font color="Green">E62</font></i></a></i>
;<br/><font color="Maroon">PF</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">G1</font></span>)</span> = 
<font color="Maroon">PF</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z0</font>
<b>by </b><i><a class="ref" href="polynom5.html#D3" target="_self">Def3</a>, <a class="txt" href="polynom5.html#E16:93"><i><font color="Green">E62</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:93_8"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">enp0</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="polynom5.html#T26" target="_self">Th26</a>, <a class="ref" href="polynom5.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div>
<a NAME="E56:93"/><b>then </b>
<font color="Maroon">enp0</font> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/>
<a NAME="E57:93"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z0</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="ref" href="polynom5.html#T24" target="_self">Th24</a>, </i>;<br/>
<a NAME="E58:93"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">z0</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="polynom5.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E59:93"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z0</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="polynom5.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E60:93"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z0</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="complfld.html#T111">COMPLFLD:111</a></i>;<br/>
<a NAME="E61:93"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z0</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="complfld.html#T111">COMPLFLD:111</a></i>;<br/>
<b>hence </b><a NAME="E62:93"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z0</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>


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