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

<b>let </b><font color="Maroon">F</font> be  <a href="rvsum_1.html#V1" target="_self">complex-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">F</font> being <a href="rvsum_1.html#V1" target="_self">complex-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>(  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 ) iff  <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> 0 );<br/>
<a NAME="E1:113"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<a NAME="E2:113"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:113_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">F</font> being <a href="rvsum_1.html#V1" target="_self">complex-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> holds <br/>(  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font> &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 ) iff  <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon">F</font> be  <a href="rvsum_1.html#V1" target="_self">complex-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E2:113_1"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<b>then consider </b><font color="Maroon">F'</font> being  <a href="rvsum_1.html#V1" target="_self">complex-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">x</font> being  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:113_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E5:113_1"/><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E6:113_1"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E6:113_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E7:113_1"/><i><font color="Green">E122</font></i>: 
 <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E8:113_1"/>
(  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 ) implies  <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:113_1_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E2:113_1_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_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="E1:113_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font> or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="rvsum_1.html#T39" target="_self">Th39</a>, <a class="ref" href="finseq_2.html#T8">FINSEQ_2:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:113_1_1_1_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:113_1_1_1_1_1"/>
 <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">F'</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:113_1_1_1_1_1"/><b>then </b>
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T34" target="_self">Th34</a>, <a class="ref" href="rvsum_1.html#T41" target="_self">Th41</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E4:113_1_1_1_1_1"/><b>then </b>
 <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rvsum_1.html#E5:113_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rvsum_1.html#T34" target="_self">Th34</a>, <a class="ref" href="rvsum_1.html#T40" target="_self">Th40</a>, <a class="ref" href="rvsum_1.html#T41" target="_self">Th41</a></i>;<br/><b>hence </b><a NAME="E5:113_1_1_1_1_1"/>
 <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T35" target="_self">Th35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:113_1_1_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:113_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rvsum_1.html#E6:113_1"><i><font color="Green">E72</font></i></a>, , , <a class="ref" href="rvsum_1.html#T40" target="_self">Th40</a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><b>hence </b><a NAME="E3:113_1_1_1_1_2"/>
 <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T35" target="_self">Th35</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:113_1_1"/>
 <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


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

<b>assume </b><a NAME="E9:113_1"/><i><font color="Green">E127</font></i>: 
 <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:113_1_2"/>
(  <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="ref" href="rvsum_1.html#T35" target="_self">Th35</a>, <a class="ref" href="rvsum_1.html#T39" target="_self">Th39</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_2_1"><b>suppose </b></a><a NAME="E1:113_1_2_1"/>
 <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:113_1_2_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font> &amp; <font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="rvsum_1.html#E5:113_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rvsum_1.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E4:113_1_2_1"/>
 <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">F'</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:113_1_2_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T34" target="_self">Th34</a>, <a class="ref" href="rvsum_1.html#T40" target="_self">Th40</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/><b>hence </b><a NAME="E6:113_1_2_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_2_2"><b>suppose </b></a><a NAME="E1:113_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:113_1_2_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="rvsum_1.html#E6:113_1"><i><font color="Green">E72</font></i></a>, , , <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><b>hence </b><a NAME="E3:113_1_2_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:113"/><i><font color="Green">E163</font></i>: 
 <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:113"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="rvsum_1.html#T33" target="_self">Th33</a>, );<br/></i>
<b>hence </b><a NAME="E5:113"/>
(  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 ) iff  <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>


</div>
