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<div class="add">

<b>set </b><font color="Maroon">c<sub>1</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>;<br/>
<a NAME="E1:64"/><i><font color="Green">E56</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1
 
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T4">COMPLFLD:4</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>( non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>  <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a>  =  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>] means ( ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="int_1.html#V1">integer</a> ) );<br/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/><a NAME="E2:64_1"/><i><font color="Green">E59</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E2:64_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 </span><b>by </b><i><a class="ref" href="uniroots.html#T1" target="_self">Th1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64"><i><font color="Green">E56</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="uniroots.html#T52" target="_self">Th52</a>, <a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="uniroots.html#T52" target="_self">Th52</a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:64_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_1_1_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="uniroots.html#T52" target="_self">Th52</a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:64_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64"><i><font color="Green">E56</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="uniroots.html#T52" target="_self">Th52</a>, <a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>, <font color="Maroon">c<sub>5</sub></font> being   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E3:64_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:64_1_1_2"/><i><font color="Green">E62</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:64_1_1_2"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( ( (  not <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><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="algseq_1.html#K5">%&gt;</a></span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>and </b><br/><a NAME="E6:64_1_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="uniroots.html#T54" target="_self">Th54</a></i>;<br/><a NAME="E7:64_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:64_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Olive">b<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 ) &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> is <a href="int_1.html#V1">integer</a> ) );<br/><a NAME="E9:64_1_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>] );<br/><a NAME="E10:64_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:64_1_1_2"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_1_1_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>6</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:64_1_1_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_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="E1:64_1_1_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/><b>thus </b><a NAME="E5:64_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="uniroots.html#E7:64_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E7:64_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T93">FINSEQ_1:93</a></i>;<br/><b>then </b><font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="uniroots.html#E7:64_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_2_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_5.html#T11">FINSEQ_5:11</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/><a NAME="E9:64_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="group_4.html#T12">GROUP_4:12</a></i>;<br/><a NAME="E10:64_1_1_2_1_1_1"/>
( 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/><a NAME="E11:64_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(1)
 <b>by </b><i><a class="txt" href="uniroots.html#E5:64_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uniroots.html#E7:64_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_2_1_1_1"><i><font color="Green">E72</font></i></a></i>;<br/><b>hence </b><a NAME="E12:64_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64"><i><font color="Green">E56</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_2_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="uniroots.html#E9:64_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="uniroots.html#T52" target="_self">Th52</a>, <a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>thus </b><a NAME="E13:64_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="int_1.html#V1">integer</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_2_1_1_1_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 </span><b>by </b><i><a class="ref" href="uniroots.html#T1" target="_self">Th1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64"><i><font color="Green">E56</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_2_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="uniroots.html#E9:64_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E11:64_1_1_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="uniroots.html#T52" target="_self">Th52</a>, <a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="uniroots.html#E9:64_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E11:64_1_1_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="uniroots.html#T52" target="_self">Th52</a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_1_2_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2_1_1_1_2_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="uniroots.html#E9:64_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E11:64_1_1_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="uniroots.html#T52" target="_self">Th52</a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:64_1_1_2_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E2:64_1_1_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:64_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E5:64_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:64_1_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:64_1_1_2_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>, <font color="Maroon">c<sub>9</sub></font> being   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E9:64_1_1_2_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E10:64_1_1_2_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E11:64_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E12:64_1_1_2_1_1_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 <b>and </b><br/><a NAME="E13:64_1_1_2_1_1_2"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E7:64_1_1_2_1_1_2"><i><font color="Green">E74</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/><b>thus </b><a NAME="E16:64_1_1_2_1_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="uniroots.html#E7:64_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E19:64_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="uniroots.html#E7:64_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="uniroots.html#E9:64_1_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_5.html#T11">FINSEQ_5:11</a></i>;<br/><a NAME="E20:64_1_1_2_1_1_2"/><b>then </b>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="group_4.html#T9">GROUP_4:9</a></i>;<br/><a NAME="E21:64_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E10:64_1_1_2_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="binop_2.html#K7">-</a> 1 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="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/><b>thus </b><a NAME="E23:64_1_1_2_1_1_2"/>
( ( <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="int_1.html#V1">integer</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_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="E1:64_1_1_2_1_1_2_1_1"/>
(  not <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or ( <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_2_1_1_1"/>
(  not <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><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="algseq_1.html#K5">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:64_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uniroots.html#E7:64_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1"><i><font color="Green">E71</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM8">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E4:64_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E81</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E5:64_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E6:64_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E21:64_1_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><a NAME="E7:64_1_1_2_1_1_2_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> 1 as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E9:64_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E10:64_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/><a NAME="E11:64_1_1_2_1_1_2_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E85</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E6:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="uniroots.html#E10:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E84</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E2:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="polynom5.html#T33">POLYNOM5:33</a></i>
;<br/><b>thus </b><a NAME="E13:64_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_2_1_1_2_1_1_1_2_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E12:64_1_1_2_1_1_2"><i><font color="Green">E77</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_1_1_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K24">*</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E12:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T6">COMPLFLD:6</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>;<br/><b>hence </b><a NAME="E3:64_1_1_2_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_1_1_2_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="xcmplx_0.html#K3">*</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E12:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T6">COMPLFLD:6</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>;<br/><b>hence </b><a NAME="E3:64_1_1_2_1_1_2_1_1_1_2_1_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM8">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E15:64_1_1_2_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E16:64_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E17:64_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>18</sub></font> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E21:64_1_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><a NAME="E18:64_1_1_2_1_1_2_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>18</sub></font> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="int_1.html#V1">integer</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_1_1_2_1_1_2_1_1_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="uniroots.html#E13:64_1_1_2_1_1_2"><i><font color="Green">E78</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_1_3_1"><b>now </b></a><div class="add"><a NAME="E1:64_1_1_2_1_1_2_1_1_1_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E2:64_1_1_2_1_1_2_1_1_1_3_1"/><b>then </b><i><font color="Green">E89</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_2_1_1_2_1_1_1_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E2:64_1_1_2_1_1_2_1_1_1_3_1"><i><font color="Green">E89</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:64_1_1_2_1_1_2_1_1_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2_1_1_2_1_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E2:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="polynom5.html#T33">POLYNOM5:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:64_1_1_2_1_1_2_1_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2_1_1_2_1_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="uniroots.html#E2:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="polynom5.html#T33">POLYNOM5:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>22</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E17:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_2_1_1_2_1_1_1_3"><i><font color="Green">E88</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font> <a href="binop_2.html#K5">*</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>22</sub></font>

;<br/>
<b>hence </b><a NAME="E7:64_1_1_2_1_1_2_1_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> is <a href="int_1.html#V1">integer</a>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E19:64_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E16:64_1_1_2_1_1_2_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="uniroots.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1)
 <b>by </b><i><a class="txt" href="uniroots.html#E5:64_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uniroots.html#E7:64_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E3:64_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/><a NAME="E4:64_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM8">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E6:64_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E83</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E7:64_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E8:64_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E21:64_1_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><a NAME="E9:64_1_1_2_1_1_2_1_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E6:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> 1 as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> 0 as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="uniroots.html#E4:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E12:64_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="uniroots.html#E6:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E13:64_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E7:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/><a NAME="E14:64_1_1_2_1_1_2_1_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E6:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E87</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E8:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="uniroots.html#E13:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E86</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><a NAME="E16:64_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="uniroots.html#E4:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E82</font></i></a></i>;<br/><b>thus </b><a NAME="E17:64_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_2_1_1_2_1_1_2_2_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E12:64_1_1_2_1_1_2"><i><font color="Green">E77</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_1_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E15:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><b>hence </b><a NAME="E3:64_1_1_2_1_1_2_1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="uniroots.html#E4:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E82</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E15:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><b>hence </b><a NAME="E3:64_1_1_2_1_1_2_1_1_2_2_1_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 <b>by </b><i><a class="txt" href="uniroots.html#E16:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E88</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM8">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E19:64_1_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E20:64_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E21:64_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>19</sub></font> holds <br/><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E21:64_1_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><a NAME="E22:64_1_1_2_1_1_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>19</sub></font> holds <br/><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="int_1.html#V1">integer</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_1_1_2_1_1_2_1_1_2_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="uniroots.html#E13:64_1_1_2_1_1_2"><i><font color="Green">E78</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>20</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uniroots.html#E2:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="uniroots.html#E4:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E82</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E21:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_2_1_1_2_1_1_2_3"><i><font color="Green">E91</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>24</sub></font> <a href="binop_2.html#K5">*</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>23</sub></font>

;<br/>
<b>hence </b><a NAME="E6:64_1_1_2_1_1_2_1_1_2_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> is <a href="int_1.html#V1">integer</a>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E23:64_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E20:64_1_1_2_1_1_2_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="uniroots.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E12:64_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="uniroots.html#E10:64_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="uniroots.html#E11:64_1_1_2"><i><font color="Green">E69</font></i></a>);<br/></i><a NAME="E13:64_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E15:64_1_1_2"/>
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E16:64_1_1_2"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(<a class="txt" href="uniroots.html#E9:64_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="uniroots.html#E13:64_1_1_2"><i><font color="Green">E70</font></i></a>);<br/></i><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>, <font color="Maroon">c<sub>8</sub></font> being   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E18:64_1_1_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>and </b><br/><a NAME="E19:64_1_1_2"/><i><font color="Green">E73</font></i>: 
( ( <font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="int_1.html#V1">integer</a> ) )
 <b>by </b><i><a class="txt" href="uniroots.html#E8:64_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="uniroots.html#E16:64_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E20:64_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:64_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uniroots.html#E7:64_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uniroots.html#E18:64_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_3.html#T55">FINSEQ_3:55</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E22:64_1_1_2"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E23:64_1_1_2"/><i><font color="Green">E76</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">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E24:64_1_1_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E6:64_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><a NAME="E25:64_1_1_2"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E26:64_1_1_2"/><i><font color="Green">E79</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E22:64_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E28:64_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="uniroots.html#E22:64_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then </b><i><font color="Green">E80</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E24:64_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="uniroots.html#E25:64_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="uniroots.html#E26:64_1_1_2"><i><font color="Green">E79</font></i></a></i>
;<br/><a NAME="E30:64_1_1_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_2_3_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E3:64_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uniroots.html#E7:64_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uniroots.html#E18:64_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="uniroots.html#E19:64_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_3.html#T55">FINSEQ_3:55</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b> <a href="binop_2.html#K7">-</a> 1 = 
<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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E1:64"><i><font color="Green">E56</font></i></a>, <a class="txt" href="uniroots.html#E23:64_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="uniroots.html#E29:64_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="uniroots.html#T42" target="_self">Th42</a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/><b>hence </b><a NAME="E3:64_1_1_2_3_1_1"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_3_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1
 ;<br/><div class="add"><b>then </b> <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> = 
<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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E1:64"><i><font color="Green">E56</font></i></a>, <a class="txt" href="uniroots.html#E23:64_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="uniroots.html#E29:64_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="uniroots.html#T42" target="_self">Th42</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41">VECTSP_1:41</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/><b>hence </b><a NAME="E3:64_1_1_2_3_1_2"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 <b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="rlvect_1.html#T31">RLVECT_1:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="int_1.html#V1">integer</a>;<br/><div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:64_1_1_2_4"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E3:64_1_1_2_4"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E4:64_1_1_2_4"/><i><font color="Green">E85</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">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E5:64_1_1_2_4"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>12</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E6:64_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><a NAME="E6:64_1_1_2_4"/><i><font color="Green">E87</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:64_1_1_2_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:64_1_1_2_4"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default">1,1 <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="graph_2.html#T9">GRAPH_2:9</a></i>;<br/><a NAME="E8:64_1_1_2_4"/><i><font color="Green">E89</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E6:64_1_1_2_4"><i><font color="Green">E87</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><a NAME="E11:64_1_1_2_4"/><i><font color="Green">E90</font></i>: 
1,1 <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="uniroots.html#E6:64_1_1_2_4"><i><font color="Green">E87</font></i></a>, <a class="ref" href="graph_2.html#T6">GRAPH_2:6</a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font>;<br/><a NAME="E12:64_1_1_2_4"/><i><font color="Green">E91</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E7:64_1_1_2_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="uniroots.html#E11:64_1_1_2_4"><i><font color="Green">E90</font></i></a>, <a class="ref" href="fvsum_1.html#T89">FVSUM_1:89</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_2_4_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_4_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2_4_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64"><i><font color="Green">E56</font></i></a>, <a class="txt" href="uniroots.html#E4:64_1_1_2_4"><i><font color="Green">E85</font></i></a>, <a class="ref" href="uniroots.html#T42" target="_self">Th42</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_4_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2_4_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2_4"><i><font color="Green">E85</font></i></a>, <a class="ref" href="uniroots.html#T42" target="_self">Th42</a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</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="64_1_1_2_4_1_1_3"><b>suppose </b></a><a NAME="E1:64_1_1_2_4_1_1_3"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2_4_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2_4"><i><font color="Green">E85</font></i></a>, <a class="ref" href="uniroots.html#T43" target="_self">Th43</a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> =  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:64_1_1_2_4_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>19</sub></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">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_2_4_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 or 1 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_2_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_4_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_4_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E3:64_1_1_2_4_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/><a NAME="E4:64_1_1_2_4_2_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_4_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:64_1_1_2_4_2_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E6:64_1_1_2_4_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> is <a href="int_1.html#V1">integer</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_2_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_4_2_1_2"/><i><font color="Green">E93</font></i>: 
1 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_4_2_1_2"/><b>then </b>
<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">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</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">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E3:64_1_1_2_4_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E4:64_1_1_2_4_2_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<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">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></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">c<sub>12</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_2_4_2_1_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_4_2_1_2_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_4_2_1_2_1_1"/><b>then </b>
<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">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:64_1_1_2_4"><i><font color="Green">E84</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_2_4_2_1_2"><i><font color="Green">E93</font></i></a></i>;<br/><a NAME="E3:64_1_1_2_4_2_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E4:64_1_1_2_4_2_1_2_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_4_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_2_4_2_1_2_1_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:64_1_1_2_4_2_1_2_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E6:64_1_1_2_4_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> is <a href="int_1.html#V1">integer</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_4_2_1_2_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:64_1_1_2_4_2_1_2_1_2"/><i><font color="Green">E96</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_4_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:64_1_1_2_4_2_1_2_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_4_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:64_1_1_2_4_2_1_2_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E6:64_1_1_2_4_2_1_2_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E2:64_1_1_2_4_2_1_2_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="uniroots.html#E5:64_1_1_2_4_2_1_2_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><i><font color="Green">E98</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_4_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="uniroots.html#E5:64_1_1_2_4_2_1_2_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E5:64_1_1_2_4_2_1_2_1_2"><i><font color="Green">E97</font></i></a></i>
;<br/><a NAME="E8:64_1_1_2_4_2_1_2_1_2"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_4_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="uniroots.html#E1:64_1_1_2_4_2_1_2_1_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:64_1_1_2_4_2_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="nat_2.html#T11">NAT_2:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="uniroots.html#E1:64_1_1_2_4"><i><font color="Green">E83</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="uniroots.html#E19:64_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E20:64_1_1_2"><i><font color="Green">E74</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>, <font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="uniroots.html#E19:64_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E20:64_1_1_2"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E14:64_1_1_2_4_2_1_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="txt" href="uniroots.html#E2:64_1_1_2_4_2_1_2_1_2"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E15:64_1_1_2_4_2_1_2_1_2"/><b>then </b>
1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:64_1_1_2_4_2_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E5:64_1_1_2_4"><i><font color="Green">E86</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T3">POLYNOM1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<font color="Maroon">c<sub>24</sub></font> <a href="binop_2.html#K5">*</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font>

;<br/><b>hence </b><a NAME="E17:64_1_1_2_4_2_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E7:64_1_1_2_4_2_1_2_1_2"><i><font color="Green">E98</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="uniroots.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E19:64_1_1_2_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K3">+</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E12:64_1_1_2_4"><i><font color="Green">E91</font></i></a>, <a class="ref" href="complfld.html#T3">COMPLFLD:3</a></i>;<br/><a NAME="E20:64_1_1_2_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="binop_2.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E21:64_1_1_2_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><i><font color="Green">E92</font></i>: <font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E5:64_1_1_2_4"><i><font color="Green">E86</font></i></a>, <a class="txt" href="uniroots.html#E8:64_1_1_2_4"><i><font color="Green">E89</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E25:64_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_2_4_4"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E3:64_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uniroots.html#E7:64_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uniroots.html#E18:64_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="uniroots.html#E19:64_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_3.html#T55">FINSEQ_3:55</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_4_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_4_4_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2_4_4_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="uniroots.html#E23:64_1_1_2_4"><i><font color="Green">E92</font></i></a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_4_4_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_4_4_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>13</sub></font> = 
<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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E23:64_1_1_2_4"><i><font color="Green">E92</font></i></a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T4">COMPLFLD:4</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41">VECTSP_1:41</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/><b>then </b> <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D10">RLVECT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
;<br/><a NAME="E4:64_1_1_2_4_4_2"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T66">VECTSP_1:66</a></i>;<br/><a NAME="E5:64_1_1_2_4_4_2"/><b>then </b>
 <a href="binop_2.html#K1">-</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/><a NAME="E6:64_1_1_2_4_4_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><b>hence </b><a NAME="E7:64_1_1_2_4_4_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>11</sub></font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E32:64_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>(<a class="txt" href="uniroots.html#E31:64_1_1_2"><i><font color="Green">E82</font></i></a>);<br/></i><b>hence </b><a NAME="E33:64_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="uniroots.html#E30:64_1_1_2"><i><font color="Green">E81</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:64"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="uniroots.html#S1" target="_self">UNIROOTS:sch 1</a>(<a class="txt" href="uniroots.html#E2:64"><i><font color="Green">E57</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:64"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( ( <span class="p1">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 ) &amp; <span class="p1">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> is <a href="int_1.html#V1">integer</a> )
 ;<br/>


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