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

<b>let </b><font color="Maroon">c<sub>1</sub></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">E66</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 ;<br/>

<a NAME="E2:93"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> =  <a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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> ] means verum;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B 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">b<sub>1</sub></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">c<sub>4</sub></font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E4:93"/><i><font color="Green">E68</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">c<sub>1</sub></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">c<sub>3</sub></font>
 
;<br/>
<a NAME="E5:93"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></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">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></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">E70</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E3:93_1"><i><font color="Green">E70</font></i></a>, <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="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></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">c<sub>3</sub></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">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></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">E71</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:93_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></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">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom5.html#E4:93"><i><font color="Green">E68</font></i></a>, <a class="txt" href="polynom5.html#E5:93"><i><font color="Green">E69</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></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">E73</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E2:93_2"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E7:93_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></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">c<sub>3</sub></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">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom5.html#E3:93_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="polynom5.html#E5:93_2"><i><font color="Green">E73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E8:93"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="cfcont_1.html#S1">CFCONT_1:sch 1</a>(<a class="txt" href="polynom5.html#E6:93"><i><font color="Green">E70</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">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">c<sub>1</sub></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#K5">-'</a> 1</span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></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">E72</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E11:93"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></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">E74</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E10:93"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:93"/><i><font color="Green">E75</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="polynom5.html#E1:93"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E14:93"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="polynom5.html#E2:93"><i><font color="Green">E67</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E15:93"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">E77</font></i>: 
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">E78</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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="txt" href="polynom5.html#E15:93"><i><font color="Green">E76</font></i></a>, <a class="ref" href="complfld.html#T96">COMPLFLD:96</a></i>;<br/>
<a NAME="E18:93"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>9</sub></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">E79</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E3:93_3"/><i><font color="Green">E80</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></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">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom5.html#E8:93"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E4:93_3"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="txt" href="polynom5.html#E2:93_3"><i><font color="Green">E79</font></i></a></i>;<br/>
<a NAME="E5:93_3"/><i><font color="Green">E82</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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E13:93"><i><font color="Green">E75</font></i></a>, <a class="ref" href="polynom5.html#T58" target="_self">Th58</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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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#E13:93"><i><font color="Green">E75</font></i></a>, <a class="ref" href="polynom5.html#T57" target="_self">Th57</a></i>;<br/>
<a NAME="E7:93_3"/><b>then </b><i><font color="Green">E83</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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>8</sub></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">E84</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>6</sub></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">c<sub>5</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:93_3_1"/><b>then </b><i><font color="Green">E85</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></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#E1:93"><i><font color="Green">E66</font></i></a>, <a class="txt" href="polynom5.html#E10:93"><i><font color="Green">E72</font></i></a>, <a class="txt" href="polynom5.html#E11:93"><i><font color="Green">E73</font></i></a>, <a class="txt" href="polynom5.html#E12:93"><i><font color="Green">E74</font></i></a>, <a class="txt" href="polynom5.html#E4:93_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="polynom5.html#E5:93_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="polynom5.html#E7:93_3"><i><font color="Green">E83</font></i></a>, <a class="ref" href="polynom5.html#T73" target="_self">Th73</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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> 0
 
<b>by </b><i><a class="ref" href="polynom5.html#T32" target="_self">Th32</a></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">c<sub>1</sub></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">c<sub>3</sub></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">c<sub>1</sub></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">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E5:93"><i><font color="Green">E69</font></i></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">c<sub>1</sub></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">c<sub>3</sub></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">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom5.html#E9:93_3"><i><font color="Green">E84</font></i></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="txt" href="polynom5.html#E3:93_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="polynom5.html#E2:93_3_1"><i><font color="Green">E85</font></i></a>, <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">c<sub>5</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>7</sub></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">c<sub>5</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:93"/><b>then </b>
<font color="Maroon">c<sub>5</sub></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">c<sub>7</sub></font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E21:93"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is    <a href="comseq_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E22:93"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font> =  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></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="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> );<br/>
<a NAME="E24:93"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>12</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E3:93_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>thus </b><a NAME="E4:93_4"/>
<font color="Maroon">c<sub>12</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E26:93"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="cfcont_1.html#S1">CFCONT_1:sch 1</a>(<a class="txt" href="polynom5.html#E24:93"><i><font color="Green">E81</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">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">c<sub>11</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E28:93"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></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">c<sub>12</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">E84</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">E85</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">c<sub>13</sub></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">E86</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E21:93"><i><font color="Green">E79</font></i></a>, <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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E35:93"/><b>then </b><i><font color="Green">E87</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E5:93"><i><font color="Green">E69</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></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">E88</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></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">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E22:93"><i><font color="Green">E80</font></i></a>, <a class="ref" href="comseq_2.html#D4">COMSEQ_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></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">c<sub>16</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></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">c<sub>17</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:93_5"/><i><font color="Green">E89</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></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">E90</font></i>: 
<font color="Maroon">c<sub>18</sub></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">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:93_5"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i><a class="ref" href="polynom5.html#T72" target="_self">Th72</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:93_5"/><i><font color="Green">E92</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E7:93_5"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>19</sub></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">c<sub>18</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></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">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:93_5"><i><font color="Green">E89</font></i></a>, <a class="txt" href="polynom5.html#E4:93_5"><i><font color="Green">E91</font></i></a>, <a class="ref" href="cfcont_1.html#T61">CFCONT_1:61</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:93_5"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E37:93"><i><font color="Green">E88</font></i></a>, <a class="txt" href="polynom5.html#E6:93_5"><i><font color="Green">E92</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E10:93_5"/>
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><a NAME="E11:93_5"/><b>then </b><i><font color="Green">E95</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">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E9:93_5"><i><font color="Green">E94</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>23</sub></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">E96</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E14:93_5"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>22</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E26:93"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E15:93_5"/><i><font color="Green">E98</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></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">E99</font></i>: <font color="Maroon">c<sub>10</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E3:93_5"><i><font color="Green">E90</font></i></a>, <a class="txt" href="polynom5.html#E13:93_5"><i><font color="Green">E96</font></i></a>, <a class="txt" href="polynom5.html#E14:93_5"><i><font color="Green">E97</font></i></a>, <a class="ref" href="polynom5.html#D12" target="_self">Def12</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E15:93_5"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E3:93_5"><i><font color="Green">E90</font></i></a>, <a class="ref" href="polynom5.html#D12" target="_self">Def12</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E15:93_5"><i><font color="Green">E98</font></i></a>, <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">c<sub>10</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E7:93_5"><i><font color="Green">E93</font></i></a>, <a class="txt" href="polynom5.html#E11:93_5"><i><font color="Green">E95</font></i></a>, <a class="txt" href="polynom5.html#E16:93_5"><i><font color="Green">E99</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E41:93"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>10</sub></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">E90</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="polynom5.html#E31:93"><i><font color="Green">E85</font></i></a></i>;<br/>
<a NAME="E43:93"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="polynom5.html#E28:93"><i><font color="Green">E83</font></i></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">E92</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>11</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="polynom5.html#E42:93"><i><font color="Green">E90</font></i></a>, <a class="ref" href="seq_2.html#T25">SEQ_2:25</a></i>;<br/>
<a NAME="E45:93"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>12</sub></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">E94</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="txt" href="polynom5.html#E30:93"><i><font color="Green">E84</font></i></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">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:93_6"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E30:93"><i><font color="Green">E84</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></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">E96</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E4:93_6"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>17</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E26:93"><i><font color="Green">E82</font></i></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">c<sub>10</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>11</sub></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">E98</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>10</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E28:93"><i><font color="Green">E83</font></i></a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span><a href="complex1.html#K17">.|</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">c<sub>17</sub></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></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></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">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E4:93_6"><i><font color="Green">E97</font></i></a></i>
;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></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">E99</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E9:93_6"/><i><font color="Green">E100</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></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">c<sub>3</sub></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">c<sub>13</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom5.html#E8:93"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E10:93_6"/>
<font color="Maroon">c<sub>13</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></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">c<sub>13</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></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">c<sub>13</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>17</sub></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">c<sub>3</sub></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">c<sub>13</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>3</sub></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">c<sub>17</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></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">c<sub>3</sub></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">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom5.html#E9:93_6"><i><font color="Green">E100</font></i></a>, <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">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></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">c<sub>17</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></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">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom5.html#E35:93"><i><font color="Green">E87</font></i></a>, <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">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>10</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E33:93"><i><font color="Green">E86</font></i></a>, <a class="txt" href="polynom5.html#E1:93_6"><i><font color="Green">E95</font></i></a>, <a class="txt" href="polynom5.html#E3:93_6"><i><font color="Green">E96</font></i></a>, <a class="txt" href="polynom5.html#E6:93_6"><i><font color="Green">E98</font></i></a>, <a class="txt" href="polynom5.html#E8:93_6"><i><font color="Green">E99</font></i></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">c<sub>10</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E44:93"><i><font color="Green">E92</font></i></a>, <a class="txt" href="polynom5.html#E45:93"><i><font color="Green">E93</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">E95</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">c<sub>10</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>11</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="txt" href="polynom5.html#E46:93"><i><font color="Green">E94</font></i></a>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/>
<i><font color="Green">E96</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">c<sub>10</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>11</sub></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">c<sub>10</sub></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">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E42:93"><i><font color="Green">E90</font></i></a>, <a class="txt" href="polynom5.html#E43:93"><i><font color="Green">E91</font></i></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">c<sub>10</sub></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">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E41:93"><i><font color="Green">E89</font></i></a>, <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">c<sub>10</sub></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="txt" href="polynom5.html#E28:93"><i><font color="Green">E83</font></i></a>, <a class="ref" href="seq_4.html#T44">SEQ_4:44</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></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">c<sub>18</sub></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">E97</font></i>: 
<font color="Maroon">c<sub>18</sub></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">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E54:93"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="cfcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i><a class="ref" href="polynom5.html#T72" target="_self">Th72</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">c<sub>19</sub></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">c<sub>19</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:93_8"/><i><font color="Green">E99</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E4:93_8"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>7</sub></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">c<sub>20</sub></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">c<sub>18</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>7</sub></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">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E54:93"><i><font color="Green">E98</font></i></a>, <a class="ref" href="cfcont_1.html#T61">CFCONT_1:61</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:93_8"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></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">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E22:93"><i><font color="Green">E80</font></i></a>, <a class="txt" href="polynom5.html#E3:93_8"><i><font color="Green">E99</font></i></a>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/><b>take </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E7:93_8"/>
<font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><a NAME="E8:93_8"/><b>then </b><i><font color="Green">E102</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">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></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">c<sub>7</sub></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">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E6:93_8"><i><font color="Green">E101</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>24</sub></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">E103</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E11:93_8"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>23</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E26:93"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E12:93_8"/><i><font color="Green">E105</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></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">E106</font></i>: <font color="Maroon">c<sub>18</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> = 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></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">c<sub>10</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E53:93"><i><font color="Green">E97</font></i></a>, <a class="txt" href="polynom5.html#E10:93_8"><i><font color="Green">E103</font></i></a>, <a class="txt" href="polynom5.html#E11:93_8"><i><font color="Green">E104</font></i></a>, <a class="ref" href="polynom5.html#D12" target="_self">Def12</a></i>
;<br/><font color="Maroon">c<sub>18</sub></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">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>18</sub></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">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E12:93_8"><i><font color="Green">E105</font></i></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E53:93"><i><font color="Green">E97</font></i></a>, <a class="ref" href="polynom5.html#D12" target="_self">Def12</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">c<sub>10</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E4:93_8"><i><font color="Green">E100</font></i></a>, <a class="txt" href="polynom5.html#E8:93_8"><i><font color="Green">E102</font></i></a>, <a class="txt" href="polynom5.html#E13:93_8"><i><font color="Green">E106</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E56:93"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E41:93"><i><font color="Green">E89</font></i></a>, <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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="txt" href="polynom5.html#E49:93"><i><font color="Green">E95</font></i></a>, <a class="txt" href="polynom5.html#E50:93"><i><font color="Green">E96</font></i></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="txt" href="polynom5.html#E13:93"><i><font color="Green">E75</font></i></a>, <a class="ref" href="polynom5.html#T59" target="_self">Th59</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="txt" href="polynom5.html#E13:93"><i><font color="Green">E75</font></i></a>, <a class="ref" href="polynom5.html#T59" target="_self">Th59</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="txt" href="polynom5.html#E16:93"><i><font color="Green">E77</font></i></a>, <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="txt" href="polynom5.html#E16:93"><i><font color="Green">E77</font></i></a>, <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E17:93"><i><font color="Green">E78</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>


</div>
