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

<b>let </b><font color="Maroon" title="c1">D</font> be   <a href="supinf_2.html#NM2" title="SUPINF_2:NM.2">Pos_Denum_Set_of_R_EAL</a>;<br/><b>let </b><font color="Maroon" title="c2">N</font> be    <a href="supinf_2.html#M1" title="SUPINF_2:mode.1">Num</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>set </b><font color="Maroon" title="c4">F</font> =  <a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>;<br/>
<a NAME="E1:83"/><i><font color="Green" title="E60">A1</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="supinf_2.html#D17" target="_self" title="SUPINF_2:def.17">Def17</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> $1 );<br/>
<a NAME="E2:83"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<a NAME="E1:83_1"/><i><font color="Green" title="E62">A3</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D17" target="_self" title="SUPINF_2:def.17">Def17</a></i>;<br/>
<a NAME="E2:83_1"/><i><font color="Green" title="E63">A4</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:83"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="supinf_2.html#T54" target="_self" title="SUPINF_2:th.54">Th54</a></i>;<br/>
<a NAME="E3:83_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 1
 
<b>by </b><i><a class="ref" href="supinf_2.html#T54" target="_self" title="SUPINF_2:th.54">Th54</a></i>;<br/>
<b>hence </b><a NAME="E4:83_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:83_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="supinf_2.html#E2:83_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="supinf_2.html#T20" target="_self" title="SUPINF_2:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:83"/><i><font color="Green" title="E62">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:83_2"/><i><font color="Green" title="E63">A6</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">k</font> )
 ;<br/>

<a NAME="E2:83_2"/><i><font color="Green" title="E64">A7</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D17" target="_self" title="SUPINF_2:def.17">Def17</a></i>;<br/>
<a NAME="E3:83_2"/><i><font color="Green" title="E65">A8</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T54" target="_self" title="SUPINF_2:th.54">Th54</a></i>;<br/>
<a NAME="E4:83_2"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:83_2"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:83_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:83_2"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="supinf_2.html#E3:83_2"><i><font color="Green" title="E65">A8</font></i></a>, <a class="ref" href="supinf_2.html#T20" target="_self" title="SUPINF_2:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:83"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="supinf_2.html#E2:83"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="supinf_2.html#E3:83"><i><font color="Green" title="E62">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:83"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K16" title="SUPINF_2:func.16">Ser</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c3">n</font> )
 ;<br/>


</div>
