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<div class="add">

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="complfld.html#K1">F_Complex</a> ;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green">E8</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>2</sub></font> holds <br/><font color="Maroon">c<sub>2</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>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> is <a href="int_1.html#V1">integer</a> ) holds <br/> <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font> is <a href="int_1.html#V1">integer</a>;<br/>
<a NAME="E2:8"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>assume </b><a NAME="E1:8_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:8_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <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>3</sub></font> &amp; not <font color="Maroon">c<sub>3</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> ) or  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>3</sub></font> is <a href="int_1.html#V1">integer</a> )
 <b>by </b><i><a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:8"/><i><font color="Green">E10</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>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</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="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:8_2"/><i><font color="Green">E12</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E3:8_2"/><i><font color="Green">E13</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>4</sub></font> holds <br/><font color="Maroon">c<sub>4</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>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM8">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> , <font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E5:8_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</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>6</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="uniroots.html#E2:8_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E5:8_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E7:8_2"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:8_2"><i><font color="Green">E12</font></i></a></i>;<br/>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:8_2_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:8_2_2"/><i><font color="Green">E18</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:8_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/><a NAME="E3:8_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:8_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="uniroots.html#E1:8_2_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E4:8_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:8_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="uniroots.html#E1:8_2_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="uniroots.html#E2:8_2_2"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</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="E10:8_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&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#E2:8_2"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E11:8_2"/><b>then </b>
0 <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="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:8_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:8_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="int_1.html#V1">integer</a>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:8_2"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="uniroots.html#E1:8_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="uniroots.html#E2:8_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="uniroots.html#E5:8_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="uniroots.html#E7:8_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="uniroots.html#E8:8_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E5:8_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#T3">COMPLFLD:3</a></i>
;<br/>
<b>hence </b><a NAME="E16:8_2"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>4</sub></font> is <a href="int_1.html#V1">integer</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:8"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <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="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="uniroots.html#E2:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="uniroots.html#E3:8"><i><font color="Green">E10</font></i></a>);<br/></i>
<a NAME="E5:8"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 
;<br/>
<b>hence </b><a NAME="E6:8"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>2</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="uniroots.html#E4:8"><i><font color="Green">E11</font></i></a></i>;<br/>


</div>
