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

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<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>set </b><font color="Maroon">p</font> =  <a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font>;<br/>
<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="E2:52"/><i><font color="Green">E31</font></i>: 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E3:52"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:52"/><i><font color="Green">E39</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>   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/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom4.html#D2">POLYNOM4:def 2</a></i>;<br/>
<a NAME="E5:52"/><i><font color="Green">E40</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:52"/><b>then </b><i><font color="Green">E41</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="uniroots.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E7:52"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="uniroots.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E8:52"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E9:52"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="uniroots.html#T3" target="_self">Th3</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>set </b><font color="Maroon">null</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>;<br/>
<a NAME="E10:52"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T132">FINSEQ_2:132</a></i>;<br/>
<a NAME="E11:52"/><i><font color="Green">E55</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="matrix_3.html#T13">MATRIX_3:13</a></i>;<br/>
<a NAME="E12:52"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E13:52"/><i><font color="Green">E58</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>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><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="E1:52_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:52_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E12:52"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E3:52_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">xn</font> = <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font>;<br/>
<b>reconsider </b><font color="Maroon">zt</font> = <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">1f</font> =  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> 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="E16:52"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
;<br/>
<a NAME="E17:52"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>set </b><font color="Maroon">tR1</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">tR2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>;<br/>
<a NAME="E18:52"/><i><font color="Green">E66</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <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="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<a NAME="E1:52_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:52_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E10:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="uniroots.html#E12:52"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>

<a NAME="E3:52_2"/>
 <a href="finseq_1.html#K3">len</a> <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="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E4:52_2"/>
 <a href="finseq_1.html#K3">len</a> <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="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E10:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="uniroots.html#E12:52"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">R1</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i>, <a class="txt" href="uniroots.html#E12:52"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_2.html#T127">FINSEQ_2:127</a></i>;<br/>
<b>reconsider </b><font color="Maroon">R2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i>, <a class="txt" href="uniroots.html#E12:52"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_2.html#T127">FINSEQ_2:127</a></i>;<br/>
<a NAME="E21:52"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> &amp; ( 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#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">R1</font> holds <br/><font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_3"/>
<font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/>

<a NAME="E2:52_3"/>
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#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">R1</font> holds <br/><font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1"><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><b>that </b><br/><a NAME="E1:52_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>and </b><br/><a NAME="E2:52_3_1"/><i><font color="Green">E69</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">R1</font>
 ;<br/>

<a NAME="E3:52_3_1"/><i><font color="Green">E70</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_3_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 ;<br/><a NAME="E2:52_3_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E3:52_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:52_3_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/>
<a NAME="E7:52_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="uniroots.html#E17:52"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:52_3_1"/>
<font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E17:52"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_3"/>
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#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">R1</font> holds <br/><font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E22:52"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font> &amp; ( 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">R2</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_4"/>
<font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E10:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="uniroots.html#E12:52"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>

<a NAME="E2:52_4"/>
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">R2</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4_1"><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="E1:52_4_1"/><i><font color="Green">E73</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">R2</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:52_4_1"/><i><font color="Green">E74</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_4_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:52_4_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="uniroots.html#E12:52"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:52_4_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:52_4_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:52_4_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E10:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:52_4_1"/><b>then </b>
<font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:52_4_1"/>
<font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="uniroots.html#E17:52"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_4"/>
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">R2</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E23:52"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <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>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E9:52"><i><font color="Green">E53</font></i></a>, <a class="ref" href="fvsum_1.html#T89">FVSUM_1:89</a></i>;<br/>
<a NAME="E24:52"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1f</font> <a href="binop_2.html#K3">+</a> <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#T3">COMPLFLD:3</a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/>
<a NAME="E25:52"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 
;<br/>
<a NAME="E26:52"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E9:52"><i><font color="Green">E53</font></i></a>, <a class="ref" href="fvsum_1.html#T87">FVSUM_1:87</a></i>;<br/>
<a NAME="E27:52"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <a href="complex1.html#K5">0c</a>  <a href="binop_2.html#K3">+</a> <font color="Maroon">zt</font>
 
<b>by </b><i><a class="ref" href="complfld.html#T3">COMPLFLD:3</a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/>
<a NAME="E28:52"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font>
 
;<br/>
<a NAME="E29:52"/><i><font color="Green">E78</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="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E30:52"/><i><font color="Green">E106</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E31:52"/><b>then </b><i><font color="Green">E170</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> &amp;  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E32:52"/><i><font color="Green">E171</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E33:52"/><b>then </b><i><font color="Green">E187</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</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="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E34:52"/>
1 <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E35:52"/><b>then </b><i><font color="Green">E188</font></i>: 
1 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E36:52"/><i><font color="Green">E205</font></i>: 
<span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E6:52"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E37:52"/>
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="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5"><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="E1:52_5"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span>
 ;<br/>

<a NAME="E2:52_5"/><i><font color="Green">E207</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="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_5"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">R1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">R2</font> )
 
<b>by </b><i>, <a class="txt" href="uniroots.html#E18:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:52_5"/><i><font color="Green">E209</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E18:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:52_5"/><i><font color="Green">E210</font></i>: 
(  <a href="binop_2.html#K1">-</a> <a href="complex1.html#K6">1r</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> &amp;  <a href="complex1.html#K6">1r</a>  <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  &amp;  <a href="complex1.html#K5">0c</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  )
 
<b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>;<br/>
<b>then </b><i><font color="Green">E211</font></i>: <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <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>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K1">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="binop_2.html#K5">*</a> <a href="complex1.html#K6">1r</a> 
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_5_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_5_2_1_1"><b>suppose </b></a><a NAME="E1:52_5_2_1_1"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E213</font></i>: <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,0</span>)</span>
<b>by </b><i>, <a class="txt" href="uniroots.html#E21:52"><i><font color="Green">E67</font></i></a>, </i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#D8">GROUP_1:def 8</a>, </i>
;<br/><a NAME="E3:52_5_2_1_1"/>
<font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i>, , , , </i>;<br/><a NAME="E4:52_5_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <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>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E13:52"><i><font color="Green">E58</font></i></a>, , , <a class="ref" href="fvsum_1.html#T21">FVSUM_1:21</a></i>;<br/><a NAME="E5:52_5_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K1">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i>, <a class="ref" href="complfld.html#T3">COMPLFLD:3</a></i>;<br/><b>hence </b><a NAME="E6:52_5_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="uniroots.html#E25:52"><i><font color="Green">E76</font></i></a>, <a class="ref" href="complfld.html#T4">COMPLFLD:4</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_5_2_1_2"><b>suppose </b></a><a NAME="E1:52_5_2_1_2"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_5_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_5_2_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_5_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_5_2_1_2_1_1_1"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><a NAME="E2:52_5_2_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="uniroots.html#T3" target="_self">Th3</a>, <a class="ref" href="uniroots.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E3:52_5_2_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E17:52"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><b>then </b><i><font color="Green">E216</font></i>: <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="txt" href="uniroots.html#E6:52"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/><a NAME="E5:52_5_2_1_2_1_1_1"/>
<font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i>, <a class="txt" href="uniroots.html#E13:52"><i><font color="Green">E58</font></i></a>, , , <a class="txt" href="uniroots.html#E25:52"><i><font color="Green">E76</font></i></a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="uniroots.html#E25:52"><i><font color="Green">E76</font></i></a>, <a class="ref" href="fvsum_1.html#T21">FVSUM_1:21</a></i>
<br/>.= 
<a href="complex1.html#K5">0c</a>  <a href="binop_2.html#K3">+</a> <font color="Maroon">zt</font>
<b>by </b><i><a class="ref" href="complfld.html#T3">COMPLFLD:3</a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font>

;<br/><b>hence </b><a NAME="E7:52_5_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E25:52"><i><font color="Green">E76</font></i></a>, <a class="txt" href="uniroots.html#E28:52"><i><font color="Green">E77</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_5_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_5_2_1_2_1_1_2"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><a NAME="E2:52_5_2_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E22:52"><i><font color="Green">E72</font></i></a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:52_5_2_1_2_1_1_2"/><b>then </b>
1 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:52_5_2_1_2_1_1_2"/><b>then </b>
1 <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K10">-</a> 1
 ;<br/><a NAME="E5:52_5_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E218</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E22:52"><i><font color="Green">E72</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_5_2_1_2_1_1_2_1"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:52_5_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E22:52"><i><font color="Green">E72</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>hence </b><a NAME="E3:52_5_2_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="uniroots.html#E25:52"><i><font color="Green">E76</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E7:52_5_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E28:52"><i><font color="Green">E77</font></i></a>, <a class="txt" href="uniroots.html#E7:52"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E8:52_5_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:52_5_2_1_2_1_1_2"/>
( <font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  &amp; <font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  )
 <b>by </b><i><a class="txt" href="uniroots.html#E13:52"><i><font color="Green">E58</font></i></a>, , , , , <a class="txt" href="uniroots.html#E25:52"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E10:52_5_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="fvsum_1.html#T21">FVSUM_1:21</a></i>;<br/><a NAME="E11:52_5_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <a href="complex1.html#K5">0c</a>  <a href="binop_2.html#K3">+</a> <a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="ref" href="complfld.html#T3">COMPLFLD:3</a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/><b>hence </b><a NAME="E12:52_5_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E29:52"><i><font color="Green">E78</font></i></a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:52_5_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:52_5"/>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E38:52"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">R1</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E17:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="uniroots.html#E18:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> =  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  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="E40:52"/><i><font color="Green">E222</font></i>: 
 <a href="binop_2.html#K1">-</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/>
<a NAME="E41:52"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <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>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="fvsum_1.html#T95">FVSUM_1:95</a></i>;<br/>
<a NAME="E42:52"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K1">-</a> <font color="Maroon">z2</font></span>)</span> <a href="binop_2.html#K3">+</a> <font color="Maroon">zt</font>
 
<b>by </b><i>, <a class="ref" href="complfld.html#T3">COMPLFLD:3</a></i>;<br/>
<b>hence </b><a NAME="E43:52"/>
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>;<br/>


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