<?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="E1:94"/><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> 1
 ;<br/>

<a NAME="E2:94"/><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#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:94_1"/>
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 or  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="ref" href="polynom5.html#T5" target="_self">Th5</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1"><b>suppose </b></a><a NAME="E1:94_1_1"/>
 <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/><div class="add"><b>then consider </b><font color="Maroon">z0</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="E3:94_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  holds  <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="Olive">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/>;<br/><b>set </b><font color="Maroon">q</font> =  <a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span>;<br/><i><font color="Green">E39</font></i>:  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z0</font> = 
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">z0</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
<br/>.= 
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:94_1_1"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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>
<b>by </b><i/>
;<br/><div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_2"><b>now </b></a><div class="add"><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/> <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span>,<font color="Maroon">z</font> = 
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">z0</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E5:94_1_1"><i><font color="Green">E40</font></i></a></i>
;<br/><a NAME="E2:94_1_1_2"/><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#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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> <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="polynom5.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E3:94_1_1_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#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E6:94_1_1"/><i><font color="Green">E45</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="E7:94_1_1"/>
 <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  <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="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/><a NAME="E8:94_1_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E46</font></i>:  <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 2
<b>by </b><i>, </i>
<br/>.= 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>

;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <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>  );<br/><a NAME="E10:94_1_1"/><i><font color="Green">E47</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:94_1_1_4"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E2:94_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</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#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<a NAME="E4:94_1_1_4"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E5:94_1_1_4"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>

<a NAME="E6:94_1_1_4"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<b>hence </b><a NAME="E7:94_1_1_4"/>
<span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <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/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">k</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="E12:94_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E13:94_1_1"/><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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="ref" href="polynom5.html#T8" target="_self">Th8</a>);<br/></i><a NAME="E14:94_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E16:94_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>;<br/><a NAME="E17:94_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>consider </b><font color="Maroon">sq</font> being    <a href="complfld.html#M1">CRoot</a> of <font color="Maroon">k1</font>, <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</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 the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">F2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E20:94_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</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#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E21:94_1_1"/><i><font color="Green">E60</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"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> holds <br/><font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</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="E22:94_1_1"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E23:94_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</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#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_5"><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/><b>assume </b><a NAME="E1:94_1_1_5"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F2</font>
 ;<br/><a NAME="E2:94_1_1_5"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:94_1_1_5"/>
<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:94_1_1_6"/><i><font color="Green">E63</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><b>set </b><font color="Maroon">z1</font> = <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font>;<br/><b>consider </b><font color="Maroon">F1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E3:94_1_1_6"/><i><font color="Green">E64</font></i>: 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F1</font>
 <b>and </b><br/><a NAME="E4:94_1_1_6"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</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#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span>
 <b>and </b><br/><a NAME="E5:94_1_1_6"/><i><font color="Green">E66</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="finsop_1.html#K5">dom</a> <font color="Maroon">F1</font> holds <br/><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom4.html#D2">POLYNOM4:def 2</a></i>;<br/><a NAME="E6:94_1_1_6"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:94_1_1_6"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:94_1_1_6"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E9:94_1_1_6"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font>
 <b>by </b><i>, , <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>;<br/><a NAME="E10:94_1_1_6"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:94_1_1_6"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E12:94_1_1_6"/><b>then </b><i><font color="Green">E132</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:94_1_1_6"/><i><font color="Green">E133</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T20">FINSEQ_5:20</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:94_1_1_6_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E2:94_1_1_6_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E3:94_1_1_6_1"/><i><font color="Green">E136</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom5.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:94_1_1_6_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E5:94_1_1_6_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="polynom5.html#E27"><i><font color="Green">Lemma93</font></i></a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:94_1_1_6_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:94_1_1_6_1"/><b>then </b>
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:94_1_1_6_1"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E9:94_1_1_6_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E10:94_1_1_6_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:94_1_1_6_1"/><b>then </b>
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E12:94_1_1_6_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T24" target="_self">Th24</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom5.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
<br/>.= 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom5.html#T21" target="_self">Th21</a>, <a class="ref" href="polynom5.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom5.html#T21" target="_self">Th21</a>, <a class="ref" href="polynom5.html#T23" target="_self">Th23</a></i>
<br/>.= 
<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="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#D1" target="_self">Def1</a>, , <a class="ref" href="polynom5.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
;<br/></div><b>end;</b></div><b>then </b><i><font color="Green">E142</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="polynom5.html#T20" target="_self">Th20</a>, <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>
<br/>.= 
<font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="polynom5.html#T20" target="_self">Th20</a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
<br/>.= 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> 1
<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="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></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></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span>,0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/><i><font color="Green">E143</font></i>: <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T100">GROUP_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D2">COMPLFLD:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="complfld.html#T78">COMPLFLD:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="complfld.html#T64">COMPLFLD:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="complfld.html#T66">COMPLFLD:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="hahnban1.html#T37">HAHNBAN1:37</a></i>
;<br/><b>set </b><font color="Maroon">gc</font> = <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span>;<br/><a NAME="E17:94_1_1_6"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E18:94_1_1_6"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E19:94_1_1_6"/>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="tbsp_1.html#T2">TBSP_1:2</a></i>;<br/><a NAME="E20:94_1_1_6"/><b>then </b><i><font color="Green">E146</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E21:94_1_1_6"/>
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/><a NAME="E22:94_1_1_6"/><b>then </b><i><font color="Green">E147</font></i>: 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></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/>;<br/><a NAME="E23:94_1_1_6"/><b>then </b><i><font color="Green">E148</font></i>: 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></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/><i><font color="Green">E149</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#D3" target="_self">Def3</a>, <a class="ref" href="complfld.html#T66">COMPLFLD:66</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#D3" target="_self">Def3</a>, <a class="ref" href="complfld.html#T64">COMPLFLD:64</a></i>
;<br/><a NAME="E25:94_1_1_6"/>
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="polynom4.html#T3">POLYNOM4:3</a></i>;<br/><b>then </b> <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F1</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T16" target="_self">Th16</a>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<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="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T43">VECTSP_1:43</a></i>
;<br/><a NAME="E27:94_1_1_6"/><b>then </b><i><font color="Green">E150</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p0">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p5"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</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/><a NAME="E28:94_1_1_6"/>
 <a href="complex1.html#K6">1r</a>  <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>;<br/><b>then </b><i><font color="Green">E151</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="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default">1,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="rlvect_1.html#K6">-</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p3">(<span class="default">0 <a href="real_1.html#K5">-</a> 0</span>)</span></span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i/>
<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E30:94_1_1_6"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p0">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p5"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p5">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="complfld.html#T100">COMPLFLD:100</a></i>;<br/><a NAME="E31:94_1_1_6"/><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"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p5">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E32:94_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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#K4">*</a> <span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p5">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="complfld.html#T109">COMPLFLD:109</a></i>;<br/><a NAME="E33:94_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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#K4">*</a> <span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p5">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span></span>)</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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="complfld.html#T109">COMPLFLD:109</a></i>;<br/><a NAME="E34:94_1_1_6"/><b>then </b>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</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="rlvect_1.html#K9">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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#K4">*</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#T25" target="_self">Th25</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E35:94_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</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="rlvect_1.html#K9">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><a NAME="E36:94_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</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="rlvect_1.html#K9">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E37:94_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p5">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#T24" target="_self">Th24</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E38:94_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p5">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</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>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E39:94_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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="rlvect_1.html#K9">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E40:94_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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="rlvect_1.html#K9">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="power.html#T34">POWER:34</a></i>;<br/><a NAME="E41:94_1_1_6"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">c</font> <a href="real_1.html#K4">*</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="rlvect_1.html#K9">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D23">VECTSP_1:def 23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T29">POLYNOM1:29</a></i>
;<br/><a NAME="E43:94_1_1_6"/><b>then </b><i><font color="Green">E154</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="rlvect_1.html#K9">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" target="_self">.|</a></span>
 <b>by </b><i/>;<br/><a NAME="E44:94_1_1_6"/><i><font color="Green">E155</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D3">POLYNOM1:def 3</a></i>;<br/><i><font color="Green">E156</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" target="_self">.|</a></span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T28" target="_self">Th28</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><font color="Maroon">F2</font></span><a href="polynom5.html#K1" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T9" target="_self">Th9</a>, , </i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_6_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:94_1_1_6_9"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" target="_self">.|</a></span>
 ;<br/><a NAME="E2:94_1_1_6_9"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" target="_self">.|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:94_1_1_6_9"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:94_1_1_6_9"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E160</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>

;<br/><a NAME="E6:94_1_1_6_9"/><i><font color="Green">E161</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><font color="Maroon">F2</font></span><a href="polynom5.html#K1" target="_self">.|</a></span>
 <b>by </b><i/>;<br/><a NAME="E7:94_1_1_6_9"/><b>then </b><i><font color="Green">E162</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><font color="Maroon">F2</font></span><a href="polynom5.html#K1" target="_self">.|</a></span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:94_1_1_6_9"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T29" target="_self">Th29</a>, <a class="ref" href="polynom5.html#T31" target="_self">Th31</a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:94_1_1_6_9"/><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <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="E10:94_1_1_6_9"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><font color="Maroon">F2</font></span><a href="polynom5.html#K1" target="_self">.|</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#T29" target="_self">Th29</a>, <a class="ref" href="polynom5.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:94_1_1_6_9"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#T9" target="_self">Th9</a>, , </i>;<br/><a NAME="E12:94_1_1_6_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E13:94_1_1_6_9"/><b>then </b><i><font color="Green">E165</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:94_1_1_6_9"/>
<font color="Maroon">i</font> <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="polynom5.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E15:94_1_1_6_9"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E16:94_1_1_6_9"/>
<font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="tbsp_1.html#T2">TBSP_1:2</a></i>;<br/><a NAME="E17:94_1_1_6_9"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="polynom5.html#T34" target="_self">Th34</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E18:94_1_1_6_9"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E19:94_1_1_6_9"/><i><font color="Green">E169</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"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><i><font color="Green">E170</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> = 
<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom5.html#T10" target="_self">Th10</a>, </i>
<br/>.= 
<font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><i><font color="Green">E171</font></i>: <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom5.html#T28" target="_self">Th28</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="polynom5.html#T28" target="_self">Th28</a>, , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">sq</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom5.html#T33" target="_self">Th33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#T100">GROUP_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><a NAME="E22:94_1_1_6_9"/><i><font color="Green">E172</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1
 ;<br/><span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom5.html#T30" target="_self">Th30</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T28" target="_self">Th28</a>, , <a class="ref" href="polynom1.html#D3">POLYNOM1:def 3</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T109">COMPLFLD:109</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="complfld.html#T110">COMPLFLD:110</a>, <a class="ref" href="polynom5.html#T40" target="_self">Th40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><font color="Maroon">c</font>,0</span><a href="hahnban1.html#K1">**]</a></span>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#T109">COMPLFLD:109</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="hahnban1.html#T37">HAHNBAN1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T36" target="_self">Th36</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p0"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p5"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">sq</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="polynom5.html#T41" target="_self">Th41</a>, <a class="ref" href="power.html#T34">POWER:34</a></i>
;<br/><a NAME="E24:94_1_1_6_9"/><b>then </b>
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <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"><font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#T35" target="_self">Th35</a>, <a class="ref" href="polynom5.html#T37" target="_self">Th37</a>, <a class="ref" href="polynom5.html#T39" target="_self">Th39</a>, <a class="ref" href="real_2.html#T147">REAL_2:147</a></i>;<br/><a NAME="E25:94_1_1_6_9"/><b>then </b>
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><font color="Maroon">F2</font></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E26:94_1_1_6_9"/>
<span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><font color="Maroon">F2</font></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T32" target="_self">Th32</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E47:94_1_1_6"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><font color="Maroon">F2</font></span><a href="polynom5.html#K1" target="_self">.|</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#T29" target="_self">Th29</a>, <a class="ref" href="integra5.html#T3">INTEGRA5:3</a></i>;<br/><a NAME="E48:94_1_1_6"/><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="rlvect_1.html#K9">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><font color="Maroon">F2</font></span><a href="polynom5.html#K1" target="_self">.|</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#T27" target="_self">Th27</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E49:94_1_1_6"/><b>then </b>
<font color="Maroon">c</font> <a href="real_1.html#K4">*</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="rlvect_1.html#K9">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="rfinseq.html#K1">/^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><font color="Maroon">F2</font></span><a href="polynom5.html#K1" target="_self">.|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#T13" target="_self">Th13</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E50:94_1_1_6"/>
<font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2"><a href="polynom5.html#K1" target="_self">|.</a><span class="default"><font color="Maroon">F2</font></span><a href="polynom5.html#K1" target="_self">.|</a></span></span>)</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="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#T26" target="_self">Th26</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E26:94_1_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#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p4"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></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#R1">&lt;=</a> 0
 <b>by </b><i/>;<br/><a NAME="E27:94_1_1"/><b>then </b><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">p</font>,<span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</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="complfld.html#T96">COMPLFLD:96</a></i>;<br/><a NAME="E28:94_1_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <font color="Olive">x</font> <a href="polynom5.html#R1" target="_self">is_a_root_of</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_7"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">z0</font>
;<br/>

<a NAME="E1:94_1_1_7"/>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z0</font> <a href="hidden.html#R1">=</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="polynom5.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>hence </b><a NAME="E2:94_1_1_7"/>
<font color="Maroon">z0</font> <a href="polynom5.html#R1" target="_self">is_a_root_of</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E6:94_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E29:94_1_1"/>
<font color="Maroon">p</font> is <a href="polynom5.html#V1" target="_self">with_roots</a>
 <b>by </b><i><a class="txt" href="polynom5.html#E9:94_1_1"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_2"><b>suppose </b></a><a NAME="E1:94_1_2"/><i><font color="Green">E174</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>set </b><font color="Maroon">np</font> =  <a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font>;<br/><a NAME="E2:94_1_2"/><i><font color="Green">E175</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="ref" href="polynom5.html#T6" target="_self">Th6</a>, <a class="txt" href="polynom5.html#E4"><i><font color="Green">Lemma52</font></i></a></i>;<br/><a NAME="E3:94_1_2"/>
 <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  <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="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/><a NAME="E4:94_1_2"/><b>then </b><i><font color="Green">E176</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</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 class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i/>;<br/><a NAME="E5:94_1_2"/><i><font color="Green">E177</font></i>: 
<span class="p1">(<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 <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="polynom5.html#T6" target="_self">Th6</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:94_1_2_1"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:94_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="ref" href="polynom5.html#T6" target="_self">Th6</a>, , , <a class="ref" href="nat_1.html#T71">NAT_1:71</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:94_1_2_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:94_1_2_1_1_1"/>
<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> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</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 class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E2:94"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:94_1_2_1_1_2"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </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> <font color="Maroon">k</font> = 
 <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i><a class="ref" href="polynom5.html#T6" target="_self">Th6</a>, , </i>
<br/>.= 
<span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</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 class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="polynom5.html#T8" target="_self">Th8</a>, <a class="txt" href="polynom5.html#E2:94"><i><font color="Green">E34</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:94_1_2"/><b>then </b><i><font color="Green">E243</font></i>: 
 <a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</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 class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#T6" target="_self">Th6</a>, , , <a class="ref" href="algseq_1.html#T28">ALGSEQ_1:28</a></i>;<br/><a NAME="E8:94_1_2"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <font color="Olive">x</font> <a href="polynom5.html#R1" target="_self">is_a_root_of</a>  <a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_2_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">z0</font> =  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<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>;<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> = 
<span class="p1">(<span class="default"><span class="p2">(<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> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z0</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/>
<b>hence </b><a NAME="E2:94_1_2_2"/>
<font color="Maroon">z0</font> <a href="polynom5.html#R1" target="_self">is_a_root_of</a>  <a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E9:94_1_2"/><b>then </b>
 <a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">p</font> is <a href="polynom5.html#V1" target="_self">with_roots</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:94_1_2"/>
<font color="Maroon">p</font> is <a href="polynom5.html#V1" target="_self">with_roots</a>
 <b>by </b><i><a class="ref" href="polynom5.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
