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

<span class="kw">set </span><font color="Maroon" title="c1">CC</font> = <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">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> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Olive" title="b1">f</font></span>)</span>;<br/>
<a NAME="E1:89"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
( <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a>  is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp;  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T67" onmouseover="rs('monoid_0/T67')" onmouseout="rh()">MONOID_0:67</a></span>;<br/>
<a NAME="E2:89"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="89_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">f</font> be    <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> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:89_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c2">f</font></span>)</span></span><br/>

<a NAME="E2:89_1"/><span class="kw">then </span>
 <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:89_1"/>
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E1:89_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="finsop_1.html#D1" onmouseover="rs('finsop_1/D1')" onmouseout="rh()">FINSOP_1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:89"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="89_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c2">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:89_2"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">i1</font> = <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">f</font> be    <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> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:89_2"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">q</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> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> , <font color="Maroon" title="c6">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> <span class="kw"> such that </span><br/><a NAME="E4:89_2"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T19" onmouseover="rs('finseq_2/T19')" onmouseout="rh()">FINSEQ_2:19</a></span>;<br/>
<a NAME="E5:89_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_2"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="finseq_2.html#T16" onmouseover="rs('finseq_2/T16')" onmouseout="rh()">FINSEQ_2:16</a></span>;<br/>
<a NAME="E6:89_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A8</font></span>: 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c5">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89_2"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E1:89_2"><span class="lab"><font color="Green" title="E49">A5</font></span></a></span>;<br/>
 <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c4">f</font> = 
<span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_2"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="flexary1.html#T46" target="_self" onmouseover="rs('flexary1/T46')" onmouseout="rh()">Th46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c6">d</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T45" target="_self" onmouseover="rs('flexary1/T45')" onmouseout="rh()">Th45</a></span>
;<br/>
<a NAME="E8:89_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A9</font></span>: 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c6">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_2.html#T31" onmouseover="rs('rvsum_2/T31')" onmouseout="rh()">RVSUM_2:31</a></span>;<br/>
<span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c4">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p2"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T3" target="_self" onmouseover="rs('flexary1/T3')" onmouseout="rh()">Th3</a>, <a class="txt" href="#E4:89_2"><span class="lab"><font color="Green" title="E51">A7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c6">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" onmouseover="rs('finsop_1/T11')" onmouseout="rh()">FINSOP_1:11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:89_2"/>
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_2.html#T32" onmouseover="rs('rvsum_2/T32')" onmouseout="rh()">RVSUM_2:32</a>, <a class="txt" href="#E6:89_2"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="#E8:89_2"><span class="lab"><font color="Green" title="E53">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E4:89"/>
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E48">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:89"/>
 for <font color="Olive" title="b1">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> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  holds   <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Olive" title="b1">f</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
