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

<b>let </b><font color="Maroon" title="c1">F</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">F</font>;<br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>
 ;<br/>

<a NAME="E2:46"/><b>then </b><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T15" target="_self" title="HALLMAR1:th.15">Th15</a></i>;<br/>
<b>defpred </b><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> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font> implies  ex <font color="Olive" title="b1">g</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,$1 st <br/>( <font color="Olive" title="b1">g</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> &amp; ( for <font color="Olive" title="b2">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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/><font color="Olive" title="b1">g</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">k</font> ) ) );<br/>
<a NAME="E3:46"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">G</font> being    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,0<b> such that </b><br/><a NAME="E3:46_1"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="hallmar1.html#T30" target="_self" title="HALLMAR1:th.30">Th30</a></i>;<br/>
<a NAME="E4:46_1"/>
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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 holds <br/><font color="Maroon" title="c3">G</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">k</font>
 
;<br/>
<b>hence </b><a NAME="E5:46_1"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,0 st <br/>( <font color="Olive" title="b1">g</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> &amp; ( for <font color="Olive" title="b2">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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 holds <br/><font color="Olive" title="b1">g</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">k</font> ) )
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:46_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:46"/><i><font color="Green" title="E40">A5</font></i>: 
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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:46_2"/><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/>

<a NAME="E2:46_2"/>
( <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font> implies  ex <font color="Olive" title="b1">g</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 st <br/>( <font color="Olive" title="b1">g</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> &amp; ( for <font color="Olive" title="b2">l</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b2">l</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#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Olive" title="b1">g</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">l</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_2_1"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_2_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font> )
 </span><b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="taxonom1.html#T1" title="TAXONOM1:th.1">TAXONOM1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_1"><b>suppose </b></a><a NAME="E1:46_2_1_1_1"/><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c4">g</font> being    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1<b> such that </b><br/><a NAME="E3:46_2_1_1_1"/><i><font color="Green" title="E44">A9</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="hallmar1.html#T30" target="_self" title="HALLMAR1:th.30">Th30</a></i>;<br/><a NAME="E4:46_2_1_1_1"/>
for <font color="Olive" title="b1">l</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</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#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Maroon" title="c4">g</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">l</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1_1_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:46_2_1_1_1"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 st <br/>( <font color="Olive" title="b1">g</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> &amp; ( for <font color="Olive" title="b2">l</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b2">l</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#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Olive" title="b1">g</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">l</font> ) )
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:46_2_1_1_1"><i><font color="Green" title="E44">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_2"><b>suppose </b></a><a NAME="E1:46_2_1_1_2"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">g</font> being    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">k</font><b> such that </b><br/><a NAME="E3:46_2_1_1_2"/><i><font color="Green" title="E44">A11</font></i>: 
( <font color="Maroon" title="c4">g</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> &amp; ( for <font color="Olive" title="b1">l</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> holds <br/><font color="Maroon" title="c4">g</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">l</font> ) )
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2"><i><font color="Green" title="E41">A6</font></i></a></i>;<br/><a NAME="E4:46_2_1_1_2"/><i><font color="Green" title="E45">A12</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="hallmar1.html#D6" target="_self" title="HALLMAR1:def.6">Def6</a></i>;<br/><a NAME="E5:46_2_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="hallmar1.html#D6" target="_self" title="HALLMAR1:def.6">Def6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">G</font> being    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1<b> such that </b><br/><a NAME="E7:46_2_1_1_2"/><i><font color="Green" title="E46">A13</font></i>: 
<font color="Maroon" title="c5">G</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:46_2_1_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="hallmar1.html#T30" target="_self" title="HALLMAR1:th.30">Th30</a></i>;<br/><a NAME="E8:46_2_1_1_2"/><i><font color="Green" title="E47">A14</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E4:46_2_1_1_2"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="hallmar1.html#D6" target="_self" title="HALLMAR1:def.6">Def6</a></i>;<br/><a NAME="E9:46_2_1_1_2"/><i><font color="Green" title="E48">A15</font></i>: 
<font color="Maroon" title="c5">G</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E7:46_2_1_1_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="hallmar1.html#T15" target="_self" title="HALLMAR1:th.15">Th15</a></i>;<br/><a NAME="E10:46_2_1_1_2"/><b>then </b><i><font color="Green" title="E49">A16</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="hallmar1.html#E8:46_2_1_1_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="hallmar1.html#T2" target="_self" title="HALLMAR1:th.2">Th2</a></i>;<br/><a NAME="E11:46_2_1_1_2"/>
<font color="Maroon" title="c4">g</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:46_2_1_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="hallmar1.html#T15" target="_self" title="HALLMAR1:th.15">Th15</a></i>;<br/><a NAME="E12:46_2_1_1_2"/><b>then </b><i><font color="Green" title="E50">A17</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="hallmar1.html#E4:46_2_1_1_2"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="hallmar1.html#T2" target="_self" title="HALLMAR1:th.2">Th2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c5">G</font> as    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <b>by </b><i><a class="txt" href="hallmar1.html#E2:46"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="hallmar1.html#E1:46_2_1_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="hallmar1.html#E10:46_2_1_1_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="hallmar1.html#T24" target="_self" title="HALLMAR1:th.24">Th24</a></i>;<br/><a NAME="E14:46_2_1_1_2"/>
for <font color="Olive" title="b1">l</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</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#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Maroon" title="c6">G</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">l</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">l</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> ;<br/>

<b>assume </b><a NAME="E1:46_2_1_1_2_1"/><i><font color="Green" title="E51">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">l</font> &amp; <font color="Maroon" title="c7">l</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#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/>

<a NAME="E2:46_2_1_1_2_1"/>
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:46_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1_1_2_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:46_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E52">A19</font></i>: 
<font color="Maroon" title="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1_1_2_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:46_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E53">A20</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="hallmar1.html#E8:46_2_1_1_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="txt" href="hallmar1.html#E9:46_2_1_1_2"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="hallmar1.html#T2" target="_self" title="HALLMAR1:th.2">Th2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_2_1_1_2_1_1"/>
( <font color="Maroon" title="c7">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> or <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1_1_2_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:46_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><a NAME="E2:46_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">l</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:46_2_1_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="hallmar1.html#E1:46_2_1_1_2_1"><i><font color="Green" title="E51">A18</font></i></a></i>;<br/><b>hence </b><a NAME="E3:46_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">G</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">l</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E2:46"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="hallmar1.html#E12:46_2_1_1_2"><i><font color="Green" title="E50">A17</font></i></a>, <a class="txt" href="hallmar1.html#E4:46_2_1_1_2_1"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="hallmar1.html#E5:46_2_1_1_2_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="hallmar1.html#T24" target="_self" title="HALLMAR1:th.24">Th24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:46_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">G</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">l</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E15:46_2_1_1_2"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 st <br/>( <font color="Olive" title="b1">g</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> &amp; ( for <font color="Olive" title="b2">l</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b2">l</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#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Olive" title="b1">g</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">l</font> ) )
 <b>by </b><i><a class="txt" href="hallmar1.html#E7:46_2_1_1_2"><i><font color="Green" title="E46">A13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:46_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:46"/>
for <font color="Olive" title="b1">n</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">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="hallmar1.html#E3:46"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="hallmar1.html#E4:46"><i><font color="Green" title="E40">A5</font></i></a>);<br/></i>
<a NAME="E6:46"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font> implies  ex <font color="Olive" title="b1">g</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">g</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> &amp; ( for <font color="Olive" title="b2">l</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Olive" title="b1">g</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">l</font> ) ) )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c3">G</font> being    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E8:46"/><i><font color="Green" title="E41">A21</font></i>: 
( <font color="Maroon" title="c3">G</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> &amp; ( for <font color="Olive" title="b1">l</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon" title="c3">G</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">l</font> ) )
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E9:46"/><i><font color="Green" title="E42">A22</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="hallmar1.html#D6" target="_self" title="HALLMAR1:def.6">Def6</a></i>;<br/>
<a NAME="E10:46"/>
for <font color="Olive" title="b1">i</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="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon" title="c3">G</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</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> ;<br/>

<b>assume </b><a NAME="E1:46_3"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:46_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E3:46_3"/>
<font color="Maroon" title="c3">G</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E8:46"><i><font color="Green" title="E41">A21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:46"/><b>then </b>
<font color="Maroon" title="c3">G</font> is    <a href="hallmar1.html#M4" title="HALLMAR1:mode.4">Singlification</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E2:46"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="hallmar1.html#E9:46"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="hallmar1.html#T27" target="_self" title="HALLMAR1:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E12:46"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="hallmar1.html#M4" title="HALLMAR1:mode.4">Singlification</a> of <font color="Maroon" title="c2">A</font> st <font color="Olive" title="b1">G</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E8:46"><i><font color="Green" title="E41">A21</font></i></a></i>;<br/>


</div>
