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

<span class="kw">set </span><font color="Maroon" title="c1">g</font> =  <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">h</font> =  <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c3">F</font></span><br/>

<a NAME="E1:97"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">F</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> $1</span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">F</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> $1</span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">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>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:97"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<a NAME="E5:97"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
( <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <font color="Maroon" title="c5">n</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="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span>) &amp; <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <font color="Maroon" title="c5">n</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="c4">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>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="setwop_2.html#D2" title="SETWOP_2:def.2">SETWOP_2:def 2</a></span>;<br/>
<a NAME="E6:97"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
 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="97_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">j</font> =  <a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">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/>
<span class="kw">set </span><font color="Maroon" title="c7">i</font> =  <a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>);<br/>
<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:97_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:97_1"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:97_1_1_1"/>
( <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c4">f</font> or  not <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c4">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:97_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c3">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="setwop_2.html#T20" title="SETWOP_2:th.20">SETWOP_2:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_1_1_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="setwop_2.html#T20" title="SETWOP_2:th.20">SETWOP_2:20</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:97_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="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:97_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
 not <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <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 class="kw">by</span> <span class="lab"><a class="ref" href="setwop_2.html#T20" title="SETWOP_2:th.20">SETWOP_2:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_1_1_1_2"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="binop_2.html#T6" title="BINOP_2:th.6">BINOP_2:6</a>, <a class="ref" href="binop_2.html#T7" title="BINOP_2:th.7">BINOP_2:7</a>, <a class="ref" href="setwop_2.html#T20" title="SETWOP_2:th.20">SETWOP_2:20</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:97_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="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:97_1"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
 not <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T8" title="FINSEQ_3:th.8">FINSEQ_3:8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">F</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span>) = 
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</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="c9">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="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T9" title="FINSEQ_1:th.9">FINSEQ_1:9</a></span>
<br/>.= 
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <font color="Maroon" title="c9">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">F</font>,<span class="p3">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">F</font>,<span class="p3">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_1"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="setwop_2.html#T2" title="SETWOP_2:th.2">SETWOP_2:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <font color="Maroon" title="c9">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="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <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="c3">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#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></span>
<br/>.= 
<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">$$</a> (<span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <font color="Maroon" title="c9">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a></span>)</span>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a></span>)</span>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E2:97_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</a></span>
<br/>.= 
<a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</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="c9">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="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_1"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="setwop_2.html#T2" title="SETWOP_2:th.2">SETWOP_2:2</a></span>
<br/>.= 
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</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="c4">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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T9" title="FINSEQ_1:th.9">FINSEQ_1:9</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:97_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:97"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
 <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>
 
;<br/>
<span class="kw">then </span><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c3">F</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a></span>)</span>)</span>)</span>) = 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#T6" title="BINOP_2:th.6">BINOP_2:6</a>, <a class="ref" href="binop_2.html#T7" title="BINOP_2:th.7">BINOP_2:7</a>, <a class="ref" href="setwiseo.html#T31" title="SETWISEO:th.31">SETWISEO:31</a></span>
<br/>.= 
<a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">finSeg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c4">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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:97"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="setwiseo.html#T31" title="SETWISEO:th.31">SETWISEO:31</a></span>
;<br/>
<a NAME="E9:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E10:97"/>
 for <font color="Olive" title="b1">k</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">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E9:97"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E6:97"><span class="lab"><font color="Green" title="E38">A3</font></span></a>);<br/></span>
<a NAME="E11:97"/><span class="kw">then </span>
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E37">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:97"/>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D13" target="_self" title="RVSUM_1:def.13">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
