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<div class="add">

<span class="kw">let </span><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> ; <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">A</font> being 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>  st <font color="Olive" title="b1">A</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>  holds <br/> ex <font color="Olive" title="b2">G</font> being   <a href="hallmar1.html#M4" title="HALLMAR1:mode.4">Singlification</a> of <font color="Olive" title="b1">A</font> st <font color="Olive" title="b2">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>  implies  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>  )</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>( $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/>
<span class="kw">assume </span><a NAME="E1:46"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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> </span><br/>

<a NAME="E2:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hallmar1.html#T15" target="_self" title="HALLMAR1:th.15">Th15</a></span>;<br/>
<a NAME="E3:46"/><span class="lab"><font color="Green" title="E39">A3</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="46_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="E1:46_1"/><span class="lab"><font color="Green" title="E40">A4</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="E2:46_1"/>
( <font color="Maroon" title="c3">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="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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_1_1"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<font color="Maroon" title="c3">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="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_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="E1:46_1_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="taxonom1.html#T1" title="TAXONOM1:th.1">TAXONOM1:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:46_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
<font color="Maroon" title="c3">k</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">  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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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> ) )</span><br/><div class="add"><span class="kw">consider </span><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#K2" title="NAT_1:func.2">+</a> 1<span class="kw"> such that </span><br/><a NAME="E3:46_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<font color="Maroon" title="c4">g</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E1:46_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="hallmar1.html#T30" target="_self" title="HALLMAR1:th.30">Th30</a></span>;<br/><a NAME="E4:46_1_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#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_1_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:46_1_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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_1_1"><span class="lab"><font color="Green" title="E43">A7</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="46_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:46_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:46_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<font color="Maroon" title="c4">g</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">and </span><br/><a NAME="E4:46_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>;<br/><a NAME="E5:46_1_1_1_2"/>
<font color="Maroon" title="c3">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="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="hallmar1.html#D6" target="_self" title="HALLMAR1:def.6">Def6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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#K2" title="NAT_1:func.2">+</a> 1<span class="kw"> such that </span><br/><a NAME="E7:46_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_1_2"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="hallmar1.html#T30" target="_self" title="HALLMAR1:th.30">Th30</a></span>;<br/><a NAME="E8:46_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_1_1_1_2"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="hallmar1.html#T15" target="_self" title="HALLMAR1:th.15">Th15</a></span>;<br/><a NAME="E9:46_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hallmar1.html#D6" target="_self" title="HALLMAR1:def.6">Def6</a></span>;<br/><a NAME="E10:46_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A14</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hallmar1.html#D6" target="_self" title="HALLMAR1:def.6">Def6</a></span>;<br/><a NAME="E11:46_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A15</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_1_2"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E8:46_1_1_1_2"><span class="lab"><font color="Green" title="E46">A12</font></span></a></span>;<br/><a NAME="E12:46_1_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_1_2"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="hallmar1.html#T15" target="_self" title="HALLMAR1:th.15">Th15</a></span>;<br/><a NAME="E13:46_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A16</font></span>: 
<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#K2" title="NAT_1:func.2">+</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E9:46_1_1_1_2"><span class="lab"><font color="Green" title="E47">A13</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="NAT_1:func.2">+</a> 1 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E1:46_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E1:46_1_1_1_2"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E11:46_1_1_1_2"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="ref" href="hallmar1.html#T24" target="_self" title="HALLMAR1:th.24">Th24</a></span>;<br/><a NAME="E15:46_1_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#K2" title="NAT_1:func.2">+</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_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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#K2" title="NAT_1:func.2">+</a> 1 implies <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> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46_1_1_1_2_1"/><span class="lab"><font color="Green" title="E51">A17</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">l</font>
 <span class="kw">and </span><br/><a NAME="E2:46_1_1_1_2_1"/><span class="lab"><font color="Green" title="E52">A18</font></span>: 
<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#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"> <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></span><br/>

<a NAME="E3:46_1_1_1_2_1"/>
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E4:46_1_1_1_2_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:46_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A19</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E6:46_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A20</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_1_1_1_2"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="txt" href="#E8:46_1_1_1_2"><span class="lab"><font color="Green" title="E46">A12</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1_2_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="E1:46_1_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#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:46_1_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><a NAME="E2:46_1_1_1_2_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_1_2"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="txt" href="#E1:46_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:46_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E1:46_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E13:46_1_1_1_2"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="#E5:46_1_1_1_2_1"><span class="lab"><font color="Green" title="E53">A19</font></span></a>, <a class="txt" href="#E6:46_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="ref" href="hallmar1.html#T24" target="_self" title="HALLMAR1:th.24">Th24</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="46_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:46_1_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#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"> <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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:46_1_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>
 ; <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="E16:46_1_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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_1_1_1_2"><span class="lab"><font color="Green" title="E45">A11</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="E3:46_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="E4:46"/><span class="lab"><font color="Green" title="E40">A21</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="46_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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="numbers.html#K6" title="NUMBERS:func.6">0</a>  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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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> ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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="numbers.html#K6" title="NUMBERS:func.6">0</a> <span class="kw"> such that </span><br/><a NAME="E3:46_2"/><span class="lab"><font color="Green" title="E41">A22</font></span>: 
<font color="Maroon" title="c3">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="hallmar1.html#T30" target="_self" title="HALLMAR1:th.30">Th30</a></span>;<br/>
<a NAME="E4:46_2"/>
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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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/>
<span class="kw">hence </span><a NAME="E5:46_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>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_2"><span class="lab"><font color="Green" title="E41">A22</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>
<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>]
 
<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="#E4:46"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E39">A3</font></span></a>);<br/></span>
<a NAME="E6:46"/><span class="kw">then </span>
(  <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/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:46"/><span class="lab"><font color="Green" title="E41">A23</font></span>: 
<font color="Maroon" title="c3">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">and </span><br/><a NAME="E9:46"/><span class="lab"><font color="Green" title="E42">A24</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<a NAME="E10:46"/><span class="lab"><font color="Green" title="E43">A25</font></span>: 
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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/>

<span class="kw">assume </span><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:46_3"/><span class="kw">then </span>
( 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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E42">A24</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>
<a NAME="E11:46"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hallmar1.html#D6" target="_self" title="HALLMAR1:def.6">Def6</a></span>;<br/>
<a NAME="E12:46"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="ref" href="hallmar1.html#T27" target="_self" title="HALLMAR1:th.27">Th27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:46"/>
 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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:46"><span class="lab"><font color="Green" title="E41">A23</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>
