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

<b>consider </b><font color="Maroon" title="c2">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E2:112_1_1"/><i><font color="Green" title="E52">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> &amp;  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="ref" href="rvsum_1.html#D14" target="_self" title="RVSUM_1:def.14">Def14</a></i>;<br/>
<a NAME="E3:112_1_1"/><i><font color="Green" title="E53">A2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E2:112_1_1"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="valued_0.html#D3" title="VALUED_0:def.3">VALUED_0:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">mc</font> =  <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> ;<br/>
<a NAME="E4:112_1_1"/><i><font color="Green" title="E54">A3</font></i>: 
<a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K4" title="FINSOP_1:func.4">findom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="setwop_2.html#D2" title="SETWOP_2:def.2">SETWOP_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:112_1_1"/><i><font color="Green" title="E55">A4</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>;<br/>
<a NAME="E7:112_1_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E8:112_1_1"/><b>then </b><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="binop_2.html#T6" title="BINOP_2:th.6">BINOP_2:6</a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">g</font> =  <a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span>;<br/>
<a NAME="E9:112_1_1"/><i><font color="Green" title="E57">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:112_1_1_1"/><i><font color="Green" title="E58">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c6">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:112_1_1_1"/>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_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="E1:112_1_1_1_1_1"/>
( <font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> or  not <font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:112_1_1_1_1_1_1"/><i><font color="Green" title="E59">A8</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 ;<br/><div class="add"><a NAME="E2:112_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="setwop_2.html#T22" title="SETWOP_2:th.22">SETWOP_2:22</a></i>;<br/><a NAME="E3:112_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E1:112_1_1_1_1_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:112_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E3:112_1_1"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:112_1_1_1_1_1_2"/>
not <font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:112_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>
 <b>by </b><i><a class="ref" href="binop_2.html#T6" title="BINOP_2:th.6">BINOP_2:6</a>, <a class="ref" href="setwop_2.html#T22" title="SETWOP_2:th.22">SETWOP_2:22</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" title="c8">a</font> = <span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>, <font color="Maroon" title="c9">b</font> = <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <font color="Maroon" title="c7">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="rvsum_1.html#E1:112_1_1_1"><i><font color="Green" title="E58">A7</font></i></a></i>;<br/>
<a NAME="E5:112_1_1_1"/>
not <font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T9" title="FINSEQ_3:th.9">FINSEQ_3:9</a></i>;<br/>
<b>then </b><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span> = 
<a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <font color="Maroon" title="c7">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p3">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c2">f</font>,<span class="p3">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="setwop_2.html#T4" title="SETWOP_2:th.4">SETWOP_2:4</a></i>
<br/>.= 
<font color="Maroon" title="c9">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E7:112_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="ref" href="finseq_1.html#T11" title="FINSEQ_1:th.11">FINSEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:112_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="rvsum_1.html#E8:112_1_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="txt" href="rvsum_1.html#E9:112_1_1"><i><font color="Green" title="E57">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E11:112_1_1"/>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c1">F</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E2:112_1_1"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="rvsum_1.html#E4:112_1_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="rvsum_1.html#E6:112_1_1"><i><font color="Green" title="E55">A4</font></i></a></i>;<br/>


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