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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">G</font> holds   <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">G</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E1:28"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:28_1"/>
for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">G</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_1_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_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="E1:28_1_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:28_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:28_1_1_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:28_1_1_1_1_1"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="28_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:28_1_1_1_1_2"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E3:28_1_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_1_1_2"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:28_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/><span class="lab"><font color="Green" title="E28">A6</font></span>:  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_7.html#T23" target="_self" title="MATRIX_7:th.23">Th23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>
;<br/><a NAME="E6:28_1_1_1_1_2"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_1_1_2"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="finseq_5.html#T24" title="FINSEQ_5:th.24">FINSEQ_5:24</a></span>;<br/><a NAME="E7:28_1_1_1_1_2"/><span class="kw">then </span>
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c5">h</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T66" title="FINSEQ_5:th.66">FINSEQ_5:66</a></span>;<br/><a NAME="E8:28_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_7.html#T22" target="_self" title="MATRIX_7:th.22">Th22</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span> = 
<font color="Maroon" title="c5">h</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28_1_1_1_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><span class="p4"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
;<br/><span class="kw">then </span> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4">(<span class="default"><span class="p5"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p3">(<span class="default"><span class="p4"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4">(<span class="default"><span class="p5"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1_1_1_1_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></span>
;<br/><span class="kw">hence </span><a NAME="E11:28_1_1_1_1_2"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:28_1_1_1_1_2"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>
<span class="kw">hence </span><a NAME="E3:28_1_1"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:28_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:28"/>
for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">G</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span><br/>

<a NAME="E2:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
;<br/>
<a NAME="E3:28_2"/><span class="kw">then </span>
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T82" title="FINSEQ_5:th.82">FINSEQ_5:82</a></span>;<br/>
<a NAME="E4:28_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_7.html#T21" target="_self" title="MATRIX_7:th.21">Th21</a></span>;<br/>
<a NAME="E5:28_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:28_2"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T11" title="GROUP_4:th.11">GROUP_4:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E4:28"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E23">A1</font></span></a>);<br/></span>
<a NAME="E5:28"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>]
 
;<br/>
<span class="kw">hence </span><a NAME="E6:28"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
