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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <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> ; <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">G</font> being  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/> for <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">G</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">I</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <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">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span> )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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">means </span>for <font color="Olive" title="b1">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><span class="p1">(<span class="default">$1 <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span>;<br/>
<a NAME="E1:39"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
for <font color="Olive" title="b1">k</font> being   <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>   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="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">k</font> be    <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> ; <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="c4">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">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="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">I</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <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="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">J</font> = <font color="Maroon" title="c5">I</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:39_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><br/>

<a NAME="E4:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T59" title="FINSEQ_3:th.59">FINSEQ_3:59</a></span>;<br/>
<a NAME="E5:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>;<br/>
<a NAME="E6:39_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</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/>
<a NAME="E7:39_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E8:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c5">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E9:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c5">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<a NAME="E11:39_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/>
<a NAME="E12:39_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a>, <a class="ref" href="finseq_2.html#T74" title="FINSEQ_2:th.74">FINSEQ_2:74</a></span>;<br/>
<a NAME="E13:39_1"/>
<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">J</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="finseq_3.html#T61" title="FINSEQ_3:th.61">FINSEQ_3:61</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c5">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">J</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="group_4.html#T25" target="_self" title="GROUP_4:th.25">Th25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="group_4.html#T28" target="_self" title="GROUP_4:th.28">Th28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T61" title="GROUP_1:th.61">GROUP_1:61</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:39_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T74" title="FINSEQ_2:th.74">FINSEQ_2:74</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="E2:39"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">I</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <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">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><br/>

<a NAME="E2:39_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon" title="c4">I</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> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
;<br/>
<a NAME="E3:39_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c2">G</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:39_2"/>
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_2"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="group_4.html#T27" target="_self" title="GROUP_4:th.27">Th27</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="E3:39"/>
for <font color="Olive" title="b1">k</font> being   <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>  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#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E18">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:39"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</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>
