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

<b>set </b><font color="Maroon">FC</font> =  <a href="complfld.html#K1">F_Complex</a> ;<br/>
<b>set </b><font color="Maroon">cFC</font> = the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>
<b>let </b><font color="Maroon">s</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/><b>let </b><font color="Maroon">r</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">c</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 ;<br/>

<a NAME="E3:7"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="uproots.html#D1">UPROOTS:def 1</a></i>;<br/>
<a NAME="E4:7"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:7"/><i><font color="Green">E40</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Olive">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font> );<br/>
<a NAME="E6:7"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">s</font></span>)</span> holds <br/> ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">s</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>;<br/>
<a NAME="E2:7_1"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  ;<br/>
<b>reconsider </b><font color="Maroon">fi</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  ;<br/>
<b>take </b>
<font color="Maroon">fi</font>
;<br/>

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

<b>thus </b><a NAME="E5:7_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E8:7"/><i><font color="Green">E53</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">s</font></span>)</span>
 <b>and </b><br/><a NAME="E9:7"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">s</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>();<br/></i>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:7_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:7_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/><b>consider </b><font color="Maroon">c1</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E4:7_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E5:7_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c1</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E6:7_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font>
 <b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:7"/><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:7"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uproots.html#K10">poly_with_roots</a> <span class="p2">(<span class="default"><font color="Maroon">s</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="uproots.html#T69">UPROOTS:69</a></i>;<br/>
<a NAME="E13:7"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> holds <br/><span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:7_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>set </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/>
<a NAME="E2:7_3"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , , <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  ;<br/>
<a NAME="E4:7_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">xc</font> = <font color="Maroon">x</font>, <font color="Maroon">cc</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E6:7_3"/><i><font color="Green">E70</font></i>: 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K1">-</a> <font color="Maroon">cc</font>
 
<b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="polynom5.html#T48">POLYNOM5:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K1">-</a> <font color="Maroon">cc</font></span>)</span> <a href="binop_2.html#K3">+</a> <font color="Maroon">xc</font>
<b>by </b><i>, <a class="ref" href="complfld.html#T3">COMPLFLD:3</a></i>
<br/>.= 
<font color="Maroon">xc</font> <a href="binop_2.html#K4">-</a> <font color="Maroon">cc</font>

<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="complfld.html#T5">COMPLFLD:5</a></i>
;<br/>
<b>hence </b><a NAME="E8:7_3"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:7"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="uproots.html#K10">poly_with_roots</a> <span class="p4">(<span class="default"><font color="Maroon">s</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/>


</div>
