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

<b>let </b><font color="Maroon">R</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">R</font> is <a href="seq_4.html#V1">bounded_above</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">rs</font> =  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">r00</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r00</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/><font color="Maroon">rs</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r00</font> ) );<br/>
<a NAME="E3:12"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:12_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E2:12_1"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<b>then consider </b><font color="Maroon">r0</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">r0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">rs</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E5:12_1"/>
for <font color="Olive">r00</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r00</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font> holds <br/><font color="Maroon">rs</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r00</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:12_1"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">r00</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r00</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> holds <br/><font color="Maroon">rs</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r00</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:12"/>
 ex <font color="Olive">s1</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="rcomp_1.html#S1">RCOMP_1:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">s1</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:12"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">r0</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">rs</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r0</font> ) )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>2</sub></font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  implies for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">n1</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>( ( <font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> ) &amp; ( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> ) ) ) &amp; ( <font color="Maroon">a<sub>2</sub></font> is  not  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 ) );<br/>
<a NAME="E7:12"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><a NAME="E1:12_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_1"/>
(  not <font color="Maroon">x</font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  or <font color="Maroon">x</font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1"><b>case </b></a><a NAME="E1:12_2_1_1_1_1"/>
<font color="Maroon">x</font> is  not  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_2_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( <font color="Maroon">x</font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  implies for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">n1</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> holds <br/>( ( <font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> ) ) ) &amp; ( <font color="Maroon">x</font> is  not  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> 1 ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_2"><b>case </b></a><a NAME="E1:12_2_1_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r10</font> = <font color="Maroon">x</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>reconsider </b><font color="Maroon">r20</font> = <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_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="E1:12_2_1_1_1_2_1_1"/>
( <font color="Maroon">r10</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r20</font> or <font color="Maroon">r10</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r20</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:12_2_1_1_1_2_1_1_1"/>
<font color="Maroon">r10</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r20</font>
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_2_1_1_1"/><b>then </b>
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">n1</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> holds <br/>( ( <font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> ) )
 ;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_2_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( <font color="Maroon">x</font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  implies for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">n1</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> holds <br/>( ( <font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> ) ) ) &amp; ( <font color="Maroon">x</font> is  not  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:12_2_1_1_1_2_1_1_2"/>
<font color="Maroon">r10</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r20</font>
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_1_2_1_1_2"/><b>then </b>
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">n1</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> holds <br/>( ( <font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> implies <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> implies <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> ) )
 ;<br/><b>hence </b><a NAME="E3:12_2_1_1_1_2_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( <font color="Maroon">x</font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  implies for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">n1</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> holds <br/>( ( <font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> ) ) ) &amp; ( <font color="Maroon">x</font> is  not  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:12_2_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( <font color="Maroon">x</font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  implies for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">n1</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> holds <br/>( ( <font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> ) ) ) &amp; ( <font color="Maroon">x</font> is  not  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> 1 ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_2_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E8:12"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><a NAME="E1:12_3"/>
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_3_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">y2</font>] )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_3_1_1_1"/>
(  not <font color="Maroon">x</font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  or <font color="Maroon">x</font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1_1_1"><b>case </b></a><a NAME="E1:12_3_1_1_1_1"/>
<font color="Maroon">x</font> is  not  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_3_1_1_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1_1_2"><b>case </b></a><a NAME="E1:12_3_1_1_1_2"/>
<font color="Maroon">x</font> is  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">x</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_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="E1:12_3_1_1_1_2_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:12_3_1_1_1_2_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><a NAME="E2:12_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:12_3_1_1_1_2_1_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:12_3_1_1_1_2_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><a NAME="E2:12_3_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:12_3_1_1_1_2_1_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:12_3_1_1_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_3_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E9:12"/>
 ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 
<b>from </b><i><a class="ref" href="recdef_1.html#S1">RECDEF_1:sch 1</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">s2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:12"/><i><font color="Green">E71</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 ;<br/>
<a NAME="E12:12"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s2</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_4"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> ;<br/>
<a NAME="E5:12_4"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E6:12_4"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_4_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_4_1_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_4_1_1_1_1"><b>case </b></a><a NAME="E1:12_4_1_1_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_4_1_1_1_1"/>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_4_1_1_1_2"><b>case </b></a><a NAME="E1:12_4_1_1_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><a NAME="E2:12_4_1_1_1_2"/><b>then </b>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:12_4_1_1_1_2"/>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:12_4_1"/>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:12_4"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E8:12_4"/><b>then </b>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<b>hence </b><a NAME="E9:12_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">s3</font> = <font color="Maroon">s2</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E14:12"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E15:12"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_5"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:12_5"/>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:12_5"/><b>then </b>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_5_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_1"><b>case </b></a><a NAME="E1:12_5_1_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_5_1_1_1"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_5_1_1_2"><b>case </b></a><a NAME="E1:12_5_1_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_5_1_1_2"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_5"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:12"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">n4</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n4</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E17:12"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">n4</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n4</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n4</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:12_6"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:12_6"/><b>then </b>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n4</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s3</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n4</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_6_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_6_1_1_1"><b>case </b></a><a NAME="E1:12_6_1_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_6_1_1_1"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_6_1_1_2"><b>case </b></a><a NAME="E1:12_6_1_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_6_1_1_2"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:12_6"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E18:12"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
;<br/>
<a NAME="E19:12"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_7"/><i><font color="Green">E109</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_7_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:12_7_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_7_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_7_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_7_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> or 0 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_7_1_1_1_1"><b>case </b></a><a NAME="E1:12_7_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:12_7_1_1_1_1"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_7_1_1_1_2"><b>case </b></a><a NAME="E1:12_7_1_1_1_2"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_7_1_1_1_2"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:12_7_1"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_7"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">n4</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n4</font> holds <br/><font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n4</font>;<br/>
<a NAME="E20:12"/>
for <font color="Olive">n4</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n4</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E21:12"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[0]
 
;<br/>
<a NAME="E22:12"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_8"/>
for <font color="Olive">n5</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n5</font> holds <br/><font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n5</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:12_8"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">S<sub>6</sub></font>[0]
 
;<br/>
<a NAME="E3:12_8"/><i><font color="Green">E116</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_8_1"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> implies <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E2:12_8_1"/>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:12_8_1"/><b>then </b>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_8_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_8_1_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_8_1_1_1_1"><b>case </b></a><a NAME="E1:12_8_1_1_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_8_1_1_1_1"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_8_1_1_1_2"><b>case </b></a><a NAME="E1:12_8_1_1_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_8_1_1_1_2"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_8_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_8_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_8_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_8_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="E1:12_8_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_8_1_2_1_1_1"><b>case </b></a><a NAME="E1:12_8_1_2_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_8_1_2_1_1_1"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="topmetr3.html#T1" target="_self">Th1</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_8_1_2_1_1_2"><b>case </b></a><a NAME="E1:12_8_1_2_1_1_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_8_1_2_1_1_2"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:12_8_1_2"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:12_8_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:12_8"/>
for <font color="Olive">n4</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">n4</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:12_8"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E23:12"/>
for <font color="Olive">m4</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">m4</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E24:12"/><b>then </b>
for <font color="Olive">m4</font>, <font color="Olive">n4</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m4</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s3</font> &amp; <font color="Olive">n4</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s3</font> &amp; <font color="Olive">m4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n4</font> holds <br/><font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n4</font>
 
;<br/>
<a NAME="E25:12"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">s3</font> is <a href="seqm_3.html#V3">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D3">SEQM_3:def 3</a></i>;<br/>
<a NAME="E26:12"/><i><font color="Green">E123</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_9"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s3</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_9"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s3</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">nx</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="topmetr3.html#T1" target="_self">Th1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>;<br/>
<a NAME="E5:12_9"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">S<sub>6</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:12_9"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_9_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:12_9_1"/>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:12_9_1"/><b>then </b>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_9_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_9_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_9_1_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_9_1_1_1_1"><b>case </b></a><a NAME="E1:12_9_1_1_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_9_1_1_1_1"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_9_1_1_1_2"><b>case </b></a><a NAME="E1:12_9_1_1_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><a NAME="E2:12_9_1_1_1_2"/><b>then </b>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:12_9_1_1_1_2"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_9_1"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:12_9"/>
for <font color="Olive">nn</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">nn</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E8:12_9"/><b>then </b>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
;<br/>
<b>hence </b><a NAME="E9:12_9"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="topmetr3.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:12"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:12_10"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s3</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:12_10"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E3:12_10"/>
 <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:12_10"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="topmetr3.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:12"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">s3</font> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>
<a NAME="E29:12"/><i><font color="Green">E132</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:12_11"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:12_11"/><b>then </b>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s3</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:12_11"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:12"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:12_12"/><i><font color="Green">E133</font></i>: 
 <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E2:12_12"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:12_12"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:12"/><b>then </b>
<font color="Maroon">s3</font> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>
<a NAME="E32:12"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">s3</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T51">SEQ_4:51</a></i>;<br/>
<a NAME="E33:12"/><b>then </b><i><font color="Green">E135</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T3">PREPOWER:3</a></i>;<br/>
<a NAME="E34:12"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:12_13"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:12_13"/><i><font color="Green">E139</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/>
<a NAME="E4:12_13"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E5:12_13"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:12_13"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E3:12"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E7:12_13"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E3:12"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>;<br/>
<a NAME="E8:12_13"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E9:12_13"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E10:12_13"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">rs</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">rs</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<a NAME="E11:12_13"/>
<font color="Maroon">rs</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:12_13"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">rs</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n2</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:12_13"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:12_13"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">rs</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n2</font>
 
<b>by </b><i><a class="txt" href="topmetr3.html#E6:12"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:12_13"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font>
 
<b>by </b><i>, <a class="ref" href="topmetr3.html#T1" target="_self">Th1</a>, <a class="ref" href="seq_4.html#T54">SEQ_4:54</a></i>;<br/>
<b>hence </b><a NAME="E16:12_13"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font>
 <b>by </b><i><a class="ref" href="topmetr3.html#T2" target="_self">Th2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_14"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_14"/><i><font color="Green">E143</font></i>: 
 <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font>
 ;<br/><b>reconsider </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E3:12_14"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="topmetr3.html#E3:12"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E4:12_14"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font>
 <b>by </b><i><a class="txt" href="topmetr3.html#E1:12"><i><font color="Green">E32</font></i></a></i>;<br/><b>hence </b><a NAME="E5:12_14"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E36:12"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E37:12"/>
 ex <font color="Olive">s</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>( <font color="Olive">s</font> is <a href="seqm_3.html#V3">non-decreasing</a> &amp; <font color="Olive">s</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , </i>;<br/>


</div>
