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

<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 implies <font color="Maroon" title="c1">p</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:95"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> </span><br/>

<a NAME="E2:95"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A2</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:95_1"/>
(  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 or  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1"><span class="kw">suppose </span></a><a NAME="E1:95_1_1"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">z0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw"> such that </span><br/><a NAME="E3:95_1_1"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  holds  <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">z</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">z0</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T74" target="_self" title="POLYNOM5:th.74">Th74</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c3">q</font> =  <a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  );<br/><a NAME="E4:95_1_1"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T41" target="_self" title="POLYNOM5:th.41">Th41</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A4</font></span>:  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&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#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="polynom5.html#T53" target="_self" title="POLYNOM5:th.53">Th53</a></span>
<br/>.= 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font>

;<br/><a NAME="E6:95_1_1"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:95_1_1_2"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E5:95_1_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E2:95_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">k</font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">k</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]</span><br/>

<a NAME="E4:95_1_1_2"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E5:95_1_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:95_1_1_2"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span><br/>

<a NAME="E6:95_1_1_2"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<span class="kw">hence </span><a NAME="E7:95_1_1_2"/>
<span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algseq_1.html#T25" title="ALGSEQ_1:th.25">ALGSEQ_1:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:95_1_1"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 <span class="kw">and </span><br/><a NAME="E9:95_1_1"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E6:95_1_1"><span class="lab"><font color="Green" title="E71">A5</font></span></a>);<br/></span><a NAME="E10:95_1_1"/><span class="lab"><font color="Green" title="E74">A8</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_1_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">k1</font> = <font color="Maroon" title="c4">k</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_1_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">sq</font> being    <a href="complfld.html#M1" title="COMPLFLD:mode.1">CRoot</a> of <font color="Maroon" title="c5">k1</font>, <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">F2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw"> such that </span><br/><a NAME="E14:95_1_1"/><span class="lab"><font color="Green" title="E75">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:95_1_1"/><span class="lab"><font color="Green" title="E76">A10</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F2</font> holds <br/><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span><a NAME="E16:95_1_1"/>
<font color="Maroon" title="c5">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_1_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="algseq_1.html#T22" title="ALGSEQ_1:th.22">ALGSEQ_1:22</a></span>;<br/><a NAME="E17:95_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E18:95_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E19:95_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:95_1_1"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><span class="lab"><font color="Green" title="E79">A13</font></span>:  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">z0</font> = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">z0</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T48" target="_self" title="POLYNOM5:th.48">Th48</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T54" target="_self" title="POLYNOM5:th.54">Th54</a></span>
;<br/><div><span class="lab"><font color="Green" title="E80">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span>,<font color="Maroon" title="c8">z</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span><br/> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span>,<font color="Maroon" title="c8">z</font> = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>,<font color="Maroon" title="c8">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T54" target="_self" title="POLYNOM5:th.54">Th54</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">z0</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T48" target="_self" title="POLYNOM5:th.48">Th48</a></span>
;<br/><a NAME="E2:95_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span>,<font color="Maroon" title="c8">z</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">z0</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1"><span class="lab"><font color="Green" title="E69">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:95_1_1_4"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span>,<font color="Maroon" title="c8">z</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:95_1_1"><span class="lab"><font color="Green" title="E79">A13</font></span></a>, <a class="ref" href="polynom5.html#T32" target="_self" title="POLYNOM5:th.32">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">c</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 implies <font color="Maroon" title="c8">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:95_1_1_5"/><span class="lab"><font color="Green" title="E81">A15</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E2:95_1_1_5"/><span class="lab"><font color="Green" title="E82">A16</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span><br/><span class="kw">set </span><font color="Maroon" title="c9">z1</font> = <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw"> such that </span><br/><a NAME="E4:95_1_1_5"/><span class="lab"><font color="Green" title="E83">A17</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c10">F1</font>
 <span class="kw">and </span><br/><a NAME="E5:95_1_1_5"/><span class="lab"><font color="Green" title="E84">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:95_1_1_5"/><span class="lab"><font color="Green" title="E85">A19</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F1</font> holds <br/><font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#D2" title="POLYNOM4:def.2">POLYNOM4:def 2</a></span>;<br/><a NAME="E7:95_1_1_5"/><span class="lab"><font color="Green" title="E86">A20</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>;<br/><a NAME="E8:95_1_1_5"/><span class="lab"><font color="Green" title="E87">A21</font></span>: 
<font color="Maroon" title="c5">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_1_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="txt" href="#E5:95_1_1_5"><span class="lab"><font color="Green" title="E84">A18</font></span></a>, <a class="ref" href="algseq_1.html#T22" title="ALGSEQ_1:th.22">ALGSEQ_1:22</a></span>;<br/><a NAME="E9:95_1_1_5"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_1_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E10:95_1_1_5"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_1_1_5"><span class="lab"><font color="Green" title="E87">A21</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/><a NAME="E11:95_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A22</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E12:95_1_1_5"/><span class="lab"><font color="Green" title="E89">A23</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T20" title="FINSEQ_5:th.20">FINSEQ_5:20</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:95_1_1_5_1"/><span class="lab"><font color="Green" title="E90">A24</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:95_1_1_5_1"/><span class="lab"><font color="Green" title="E91">A25</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span><br/><a NAME="E3:95_1_1_5_1"/><span class="lab"><font color="Green" title="E92">A26</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5_1"><span class="lab"><font color="Green" title="E90">A24</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:95_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95_1_1_5_1"><span class="lab"><font color="Green" title="E91">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:95_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:95_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E7:95_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A27</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E8:95_1_1_5_1"/>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5_1"><span class="lab"><font color="Green" title="E90">A24</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E9:95_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_1_1_5"><span class="lab"><font color="Green" title="E87">A21</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/><a NAME="E10:95_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:95_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A28</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E12:95_1_1_5_1"/>
<font color="Maroon" title="c11">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_5_1"><span class="lab"><font color="Green" title="E92">A26</font></span></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E13:95_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A29</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:95_1_1_5_1"><span class="lab"><font color="Green" title="E94">A28</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font> = 
<font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5_1"><span class="lab"><font color="Green" title="E90">A24</font></span></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></span>
<br/>.= 
<font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:95_1_1_5"><span class="lab"><font color="Green" title="E89">A23</font></span></a>, <a class="txt" href="#E1:95_1_1_5_1"><span class="lab"><font color="Green" title="E90">A24</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95_1_1_5"><span class="lab"><font color="Green" title="E85">A19</font></span></a>, <a class="txt" href="#E12:95_1_1_5"><span class="lab"><font color="Green" title="E89">A23</font></span></a>, <a class="txt" href="#E1:95_1_1_5_1"><span class="lab"><font color="Green" title="E90">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:95_1_1"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="txt" href="#E7:95_1_1_5_1"><span class="lab"><font color="Green" title="E93">A27</font></span></a>, <a class="txt" href="#E13:95_1_1_5_1"><span class="lab"><font color="Green" title="E95">A29</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A30</font></span>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:95_1_1_5"><span class="lab"><font color="Green" title="E88">A22</font></span></a>, <a class="ref" href="polynom2.html#T5" title="POLYNOM2:th.5">POLYNOM2:5</a></span>
<br/>.= 
<font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:95_1_1_5"><span class="lab"><font color="Green" title="E88">A22</font></span></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></span>
<br/>.= 
<font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:95_1_1_5"><span class="lab"><font color="Green" title="E88">A22</font></span></a>, <a class="txt" href="#E12:95_1_1_5"><span class="lab"><font color="Green" title="E89">A23</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95_1_1_5"><span class="lab"><font color="Green" title="E85">A19</font></span></a>, <a class="txt" href="#E11:95_1_1_5"><span class="lab"><font color="Green" title="E88">A22</font></span></a>, <a class="txt" href="#E12:95_1_1_5"><span class="lab"><font color="Green" title="E89">A23</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D8" title="GROUP_1:def.8">GROUP_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
;<br/><a NAME="E15:95_1_1_5"/>
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:95_1_1"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E17:95_1_1"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="txt" href="#E5:95_1_1_5"><span class="lab"><font color="Green" title="E84">A18</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E16:95_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A31</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E17:95_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A32</font></span>: 
<font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">gc</font> = <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>;<br/><a NAME="E18:95_1_1_5"/><span class="lab"><font color="Green" title="E93">A33</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A34</font></span>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a>, <a class="ref" href="complfld.html#T66" title="COMPLFLD:th.66">COMPLFLD:66</a></span>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:95_1_1_5"><span class="lab"><font color="Green" title="E93">A33</font></span></a>, <a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a>, <a class="ref" href="complfld.html#T64" title="COMPLFLD:th.64">COMPLFLD:64</a></span>
;<br/><span class="lab"><font color="Green" title="E95">A35</font></span>: <font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:95_1_1_5"><span class="lab"><font color="Green" title="E91">A31</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95_1_1_5"><span class="lab"><font color="Green" title="E85">A19</font></span></a>, <a class="txt" href="#E16:95_1_1_5"><span class="lab"><font color="Green" title="E91">A31</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">sq</font></span>)</span>,<font color="Maroon" title="c5">k1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<font color="Maroon" title="c5">k1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<font color="Maroon" title="c5">k1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T100" title="GROUP_1:th.100">GROUP_1:100</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<font color="Maroon" title="c5">k1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D2" title="COMPLFLD:def.2">COMPLFLD:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<font color="Maroon" title="c5">k1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<font color="Maroon" title="c5">k1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_1_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="complfld.html#T78" title="COMPLFLD:th.78">COMPLFLD:78</a></span>
<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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<font color="Maroon" title="c5">k1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_1_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="complfld.html#T64" title="COMPLFLD:th.64">COMPLFLD:64</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<font color="Maroon" title="c5">k1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_1_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="complfld.html#T66" title="COMPLFLD:th.66">COMPLFLD:66</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="hahnban1.html#T37" title="HAHNBAN1:th.37">HAHNBAN1:37</a></span>
;<br/><a NAME="E21:95_1_1_5"/><span class="lab"><font color="Green" title="E96">A36</font></span>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p5"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T100" title="COMPLFLD:th.100">COMPLFLD:100</a></span>;<br/><a NAME="E22:95_1_1_5"/>
<font color="Maroon" title="c10">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:95_1_1"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E17:95_1_1"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="txt" href="#E5:95_1_1_5"><span class="lab"><font color="Green" title="E84">A18</font></span></a>, <a class="ref" href="polynom4.html#T3" title="POLYNOM4:th.3">POLYNOM4:3</a></span>;<br/><span class="kw">then </span> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c10">F1</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:95_1_1_5"><span class="lab"><font color="Green" title="E92">A32</font></span></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:95_1_1_5"><span class="lab"><font color="Green" title="E90">A30</font></span></a>, <a class="txt" href="#E20:95_1_1_5"><span class="lab"><font color="Green" title="E95">A35</font></span></a>, <a class="ref" href="rlvect_1.html#T61" title="RLVECT_1:th.61">RLVECT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></span>
<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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:95_1_1_5"><span class="lab"><font color="Green" title="E94">A34</font></span></a>, <a class="ref" href="vectsp_1.html#T43" title="VECTSP_1:th.43">VECTSP_1:43</a></span>
;<br/><a NAME="E24:95_1_1_5"/><span class="kw">then </span>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p5"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:95_1_1"><span class="lab"><font color="Green" title="E80">A14</font></span></a>, <a class="txt" href="#E4:95_1_1_5"><span class="lab"><font color="Green" title="E83">A17</font></span></a></span>;<br/><a NAME="E25:95_1_1_5"/><span class="kw">then </span>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:95_1_1_5"><span class="lab"><font color="Green" title="E96">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E26:95_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T109" title="COMPLFLD:th.109">COMPLFLD:109</a></span>;<br/><a NAME="E27:95_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A37</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T109" title="COMPLFLD:th.109">COMPLFLD:109</a></span>;<br/><a NAME="E28:95_1_1_5"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="txt" href="#E2:95_1_1_5"><span class="lab"><font color="Green" title="E82">A16</font></span></a>, <a class="ref" href="tbsp_1.html#T2" title="TBSP_1:th.2">TBSP_1:2</a></span>;<br/><a NAME="E29:95_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A38</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E30:95_1_1_5"/><span class="lab"><font color="Green" title="E99">A39</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></span>;<br/><span class="lab"><font color="Green" title="E100">A40</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:95_1_1_5"><span class="lab"><font color="Green" title="E86">A20</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:95_1_1"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="txt" href="#E5:95_1_1_5"><span class="lab"><font color="Green" title="E84">A18</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:95_1_1"><span class="lab"><font color="Green" title="E78">A12</font></span></a>, <a class="txt" href="#E5:95_1_1_5"><span class="lab"><font color="Green" title="E84">A18</font></span></a>, <a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_5_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> implies <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )</span><br/><span class="lab"><font color="Green" title="E101">A41</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font>

;<br/><span class="kw">assume </span><a NAME="E2:95_1_1_5_7"/><span class="lab"><font color="Green" title="E102">A42</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/><a NAME="E3:95_1_1_5_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A43</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:95_1_1_5_7"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:95_1_1_5"><span class="lab"><font color="Green" title="E100">A40</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E5:95_1_1_5_7"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:95_1_1"><span class="lab"><font color="Green" title="E78">A12</font></span></a>, <a class="txt" href="#E5:95_1_1_5"><span class="lab"><font color="Green" title="E84">A18</font></span></a>, <a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></span>;<br/><a NAME="E6:95_1_1_5_7"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E7:95_1_1_5_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A44</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:95_1_1_5_7"/>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_5_7"><span class="lab"><font color="Green" title="E103">A43</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E9:95_1_1_5_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A45</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E10:95_1_1_5_7"/>
<font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="txt" href="#E2:95_1_1_5"><span class="lab"><font color="Green" title="E82">A16</font></span></a>, <a class="ref" href="tbsp_1.html#T2" title="TBSP_1:th.2">TBSP_1:2</a></span>;<br/><a NAME="E11:95_1_1_5_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A46</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:95_1_1_5_7"><span class="lab"><font color="Green" title="E105">A45</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E12:95_1_1_5_7"/><span class="lab"><font color="Green" title="E107">A47</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></span>;<br/><a NAME="E13:95_1_1_5_7"/><span class="lab"><font color="Green" title="E108">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E14:95_1_1_5_7"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_5_7"><span class="lab"><font color="Green" title="E103">A43</font></span></a>, <a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></span>;<br/><a NAME="E15:95_1_1_5_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A49</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A50</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:95_1_1_5"><span class="lab"><font color="Green" title="E86">A20</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:95_1_1"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="txt" href="#E5:95_1_1_5"><span class="lab"><font color="Green" title="E84">A18</font></span></a>, <a class="txt" href="#E7:95_1_1_5"><span class="lab"><font color="Green" title="E86">A20</font></span></a>, <a class="txt" href="#E15:95_1_1_5_7"><span class="lab"><font color="Green" title="E109">A49</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">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" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95_1_1_5"><span class="lab"><font color="Green" title="E85">A19</font></span></a>, <a class="txt" href="#E7:95_1_1_5_7"><span class="lab"><font color="Green" title="E104">A44</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5_7"><span class="lab"><font color="Green" title="E101">A41</font></span></a>, <a class="ref" href="group_1.html#T100" title="GROUP_1:th.100">GROUP_1:100</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>

;<br/><a NAME="E17:95_1_1_5_7"/><span class="lab"><font color="Green" title="E111">A51</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></span>;<br/><a NAME="E18:95_1_1_5_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A52</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><span class="lab"><font color="Green" title="E113">A53</font></span>: <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95_1_1_5_7"><span class="lab"><font color="Green" title="E102">A42</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:95_1_1_5_7"><span class="lab"><font color="Green" title="E109">A49</font></span></a>, <a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></span>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:95_1_1_5"><span class="lab"><font color="Green" title="E86">A20</font></span></a>, <a class="txt" href="#E15:95_1_1_5_7"><span class="lab"><font color="Green" title="E109">A49</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T109" title="COMPLFLD:th.109">COMPLFLD:109</a></span>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:95_1_1_5"><span class="lab"><font color="Green" title="E93">A33</font></span></a>, <a class="txt" href="#E16:95_1_1_5_7"><span class="lab"><font color="Green" title="E110">A50</font></span></a>, <a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a>, <a class="ref" href="complfld.html#T110" title="COMPLFLD:th.110">COMPLFLD:110</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c8">c</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T109" title="COMPLFLD:th.109">COMPLFLD:109</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="hahnban1.html#T37" title="HAHNBAN1:th.37">HAHNBAN1:37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:95_1_1_5_7"><span class="lab"><font color="Green" title="E107">A47</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p0"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:95_1_1_5"><span class="lab"><font color="Green" title="E93">A33</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</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" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="txt" href="#E13:95_1_1_5_7"><span class="lab"><font color="Green" title="E108">A48</font></span></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>
;<br/><a NAME="E20:95_1_1_5_7"/><span class="lab"><font color="Green" title="E114">A54</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:95_1_1_5"><span class="lab"><font color="Green" title="E100">A40</font></span></a>, <a class="txt" href="#E3:95_1_1_5_7"><span class="lab"><font color="Green" title="E103">A43</font></span></a>, <a class="txt" href="#E17:95_1_1_5_7"><span class="lab"><font color="Green" title="E111">A51</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">sq</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:95_1_1"><span class="lab"><font color="Green" title="E76">A10</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:95_1_1_5_7"><span class="lab"><font color="Green" title="E114">A54</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/><a NAME="E22:95_1_1_5_7"/><span class="kw">then </span>
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:95_1_1_5_7"><span class="lab"><font color="Green" title="E106">A46</font></span></a>, <a class="txt" href="#E19:95_1_1_5_7"><span class="lab"><font color="Green" title="E113">A53</font></span></a>, <a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a>, <a class="ref" href="xreal_1.html#T155" title="XREAL_1:th.155">XREAL_1:155</a></span>;<br/><a NAME="E23:95_1_1_5_7"/><span class="kw">then </span>
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:95_1_1_5_7"><span class="lab"><font color="Green" title="E114">A54</font></span></a>, <a class="ref" href="polynom5.html#D1" target="_self" title="POLYNOM5:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E24:95_1_1_5_7"/>
<span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:95_1_1_5_7"><span class="lab"><font color="Green" title="E112">A52</font></span></a>, <a class="txt" href="#E20:95_1_1_5_7"><span class="lab"><font color="Green" title="E114">A54</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E33:95_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A55</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:95_1_1_5"><span class="lab"><font color="Green" title="E100">A40</font></span></a>, <a class="ref" href="integra5.html#T3" title="INTEGRA5:th.3">INTEGRA5:3</a></span>;<br/><span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a></span>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T6" target="_self" title="POLYNOM5:th.6">Th6</a></span>
<br/>.= 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:95_1_1_5"><span class="lab"><font color="Green" title="E98">A38</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/><a NAME="E35:95_1_1_5"/><span class="kw">then </span>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:95_1_1_5"><span class="lab"><font color="Green" title="E97">A37</font></span></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/><a NAME="E36:95_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:95_1_1_5"><span class="lab"><font color="Green" title="E93">A33</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E37:95_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p5"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p0">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:95_1_1_5"><span class="lab"><font color="Green" title="E99">A39</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E38:95_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:95_1_1_5"><span class="lab"><font color="Green" title="E99">A39</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/><a NAME="E39:95_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E40:95_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A56</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></span>;<br/><span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D23" title="VECTSP_1:def.23">VECTSP_1:def 23</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T29" title="POLYNOM1:th.29">POLYNOM1:29</a></span>
;<br/><a NAME="E42:95_1_1_5"/><span class="kw">then </span>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T15" target="_self" title="POLYNOM5:th.15">Th15</a></span>;<br/><a NAME="E43:95_1_1_5"/><span class="kw">then </span>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:95_1_1_5"><span class="lab"><font color="Green" title="E101">A55</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E44:95_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_5"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><span class="kw">hence </span><a NAME="E45:95_1_1_5"/>
<font color="Maroon" title="c8">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2"><a href="polynom5.html#K1" title="POLYNOM5:func.1">|.</a><span class="default"><font color="Maroon" title="c7">F2</font></span><a href="polynom5.html#K1" title="POLYNOM5:func.1">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:95_1_1_5"><span class="lab"><font color="Green" title="E102">A56</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E23:95_1_1"/><span class="kw">then </span>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="polynom5.html#E4"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a></span>;<br/><a NAME="E24:95_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A57</font></span>: 
<span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c1">p</font>,<span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font>,<span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T96" title="COMPLFLD:th.96">COMPLFLD:96</a></span>;<br/><a NAME="E25:95_1_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st <font color="Olive" title="b1">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c1">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c2">z0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z0</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c1">p</font></span><br/>

<a NAME="E1:95_1_1_6"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:95_1_1"><span class="lab"><font color="Green" title="E79">A13</font></span></a>, <a class="txt" href="#E24:95_1_1"><span class="lab"><font color="Green" title="E81">A57</font></span></a>, <a class="ref" href="polynom5.html#T32" target="_self" title="POLYNOM5:th.32">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:95_1_1_6"/>
<font color="Maroon" title="c2">z0</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D6" target="_self" title="POLYNOM5:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E26:95_1_1"/>
<font color="Maroon" title="c1">p</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D7" target="_self" title="POLYNOM5:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2"><span class="kw">suppose </span></a><a NAME="E1:95_1_2"/><span class="lab"><font color="Green" title="E69">A58</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c2">np</font> =  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font>;<br/><a NAME="E2:95_1_2"/><span class="lab"><font color="Green" title="E70">A59</font></span>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_2"><span class="lab"><font color="Green" title="E69">A58</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E3:95_1_2"/><span class="lab"><font color="Green" title="E71">A60</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_2"><span class="lab"><font color="Green" title="E69">A58</font></span></a>, <a class="ref" href="polynom5.html#T58" target="_self" title="POLYNOM5:th.58">Th58</a></span>;<br/><div><span class="lab"><font color="Green" title="E72">A61</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> implies <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:95_1_2_1"/><span class="lab"><font color="Green" title="E73">A62</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:95_1_2_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_2"><span class="lab"><font color="Green" title="E69">A58</font></span></a>, <a class="txt" href="#E3:95_1_2"><span class="lab"><font color="Green" title="E71">A60</font></span></a>, <a class="txt" href="#E1:95_1_2_1"><span class="lab"><font color="Green" title="E73">A62</font></span></a>, <a class="ref" href="nat_1.html#T23" title="NAT_1:th.23">NAT_1:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:95_1_2_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:95_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T39" target="_self" title="POLYNOM5:th.39">Th39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:95_1_2_1_1_2"/><span class="lab"><font color="Green" title="E74">A63</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_2"><span class="lab"><font color="Green" title="E69">A58</font></span></a>, <a class="txt" href="#E2:95_1_2"><span class="lab"><font color="Green" title="E70">A59</font></span></a>, <a class="ref" href="polynom5.html#T57" target="_self" title="POLYNOM5:th.57">Th57</a></span>
<br/>.= 
<span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A63</font></span></a>, <a class="ref" href="polynom5.html#T39" target="_self" title="POLYNOM5:th.39">Th39</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E5:95_1_2"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T41" target="_self" title="POLYNOM5:th.41">Th41</a></span>;<br/><a NAME="E6:95_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A64</font></span>: 
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_2"><span class="lab"><font color="Green" title="E69">A58</font></span></a>, <a class="txt" href="#E4:95_1_2"><span class="lab"><font color="Green" title="E72">A61</font></span></a>, <a class="ref" href="polynom5.html#T58" target="_self" title="POLYNOM5:th.58">Th58</a>, <a class="ref" href="algseq_1.html#T28" title="ALGSEQ_1:th.28">ALGSEQ_1:28</a></span>;<br/><a NAME="E7:95_1_2"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st <font color="Olive" title="b1">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c3">z0</font> =  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z0</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span><br/>

 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">z0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">z0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95_1_2"><span class="lab"><font color="Green" title="E73">A64</font></span></a>, <a class="ref" href="polynom5.html#T48" target="_self" title="POLYNOM5:th.48">Th48</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:95_1_2_2"/>
<font color="Maroon" title="c3">z0</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D6" target="_self" title="POLYNOM5:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:95_1_2"/><span class="kw">then </span>
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c1">p</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D7" target="_self" title="POLYNOM5:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E9:95_1_2"/>
<font color="Maroon" title="c1">p</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_2"><span class="lab"><font color="Green" title="E69">A58</font></span></a>, <a class="ref" href="polynom5.html#T61" target="_self" title="POLYNOM5:th.61">Th61</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
