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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">N</font> being    <a href="supinf_2.html#M1" title="SUPINF_2:mode.1">Num</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b2">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 <br/> ( <span class="p1">(<span class="default"><a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Olive" title="b1">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Olive" title="b1">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Olive" title="b1">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">n</font> )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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 <br/> ( <span class="p1">(<span class="default"><a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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> &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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">n</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">F</font> =  <a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>);<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span>( <span class="p1">(<span class="default"><a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1 );<br/>
<a NAME="E1:51"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def17</a></span>;<br/>
<a NAME="E2:51"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 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="51_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <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] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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>
 <span class="kw">and </span><br/><a NAME="E2:51_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 <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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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]</span><br/>

<a NAME="E3:51_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <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#K12" title="SUPINF_2:func.12">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T35" target="_self" title="SUPINF_2:th.35">Th54</a></span>;<br/>
<a NAME="E4:51_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:51_1"/>
<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]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xxreal_3.html#T39" title="XXREAL_3:th.39">XXREAL_3:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:51"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def17</a></span>;<br/>
<a NAME="E4:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="supinf_2.html#T35" target="_self" title="SUPINF_2:th.35">Th54</a>, <a class="ref" href="xxreal_3.html#T39" title="XXREAL_3:th.39">XXREAL_3:39</a></span>;<br/>
<a NAME="E5:51"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E32">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:51"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K15" title="SUPINF_2:func.15">Ser</a> (<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">N</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
