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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 ;<br/>

<a NAME="E2:46"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T15" target="_self">Th15</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) ) );<br/>
<a NAME="E3:46"/><i><font color="Green">E39</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">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,0<b> such that </b><br/><a NAME="E3:46_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46"><i><font color="Green">E37</font></i></a>, <a class="ref" href="hallmar1.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E4:46_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 holds <br/><font color="Maroon">c<sub>3</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E5:46_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,0 st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 holds <br/><font color="Olive">b<sub>1</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:46_1"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:46"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<a NAME="E2:46_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Olive">b<sub>1</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></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">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="taxonom1.html#T1">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">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1<b> such that </b><br/><a NAME="E3:46_2_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46"><i><font color="Green">E37</font></i></a>, <a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="hallmar1.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E4:46_2_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">c<sub>4</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:46_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Olive">b<sub>1</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:46_2_1_1_1"><i><font color="Green">E44</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">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:46_2_1_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E4:46_2_1_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="hallmar1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E5:46_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="hallmar1.html#D6" target="_self">Def6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1<b> such that </b><br/><a NAME="E7:46_2_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:46_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="hallmar1.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E8:46_2_1_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E4:46_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="hallmar1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E9:46_2_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E7:46_2_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="hallmar1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E10:46_2_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="hallmar1.html#E8:46_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="hallmar1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E11:46_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:46_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="hallmar1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E12:46_2_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="hallmar1.html#E4:46_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="hallmar1.html#T2" target="_self">Th2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <b>by </b><i><a class="txt" href="hallmar1.html#E2:46"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="hallmar1.html#E1:46_2_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="hallmar1.html#E10:46_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="hallmar1.html#E12:46_2_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="hallmar1.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E14:46_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">c<sub>6</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_2_1_1_2_1"/><i><font color="Green">E51</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E2:46_2_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:46_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1_1_2_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:46_2_1_1_2_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1_1_2_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:46_2_1_1_2_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="hallmar1.html#E8:46_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hallmar1.html#E9:46_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="hallmar1.html#T2" target="_self">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">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="hallmar1.html#E1:46_2_1_1_2_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</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">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:46_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:46_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hallmar1.html#E1:46_2_1_1_2_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E3:46_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E2:46"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hallmar1.html#E1:46_2_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="hallmar1.html#E12:46_2_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="hallmar1.html#E4:46_2_1_1_2_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="hallmar1.html#E5:46_2_1_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="hallmar1.html#T24" target="_self">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">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></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">b<sub>1</sub></font> being   <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Olive">b<sub>1</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="hallmar1.html#E7:46_2_1_1_2"><i><font color="Green">E46</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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:46"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="hallmar1.html#E3:46"><i><font color="Green">E39</font></i></a>, <a class="txt" href="hallmar1.html#E4:46"><i><font color="Green">E40</font></i></a>);<br/></i>
<a NAME="E6:46"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) ) )
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:46"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="hallmar1.html#E6:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E9:46"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="hallmar1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E10:46"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:46_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E3:46_3"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E8:46"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:46"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is    <a href="hallmar1.html#M4" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E2:46"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hallmar1.html#E9:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="hallmar1.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E12:46"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hallmar1.html#M4" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 <b>by </b><i><a class="txt" href="hallmar1.html#E8:46"><i><font color="Green">E42</font></i></a></i>;<br/>


</div>
