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

<b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:102"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">F</font> is <a href="supinf_2.html#V6" target="_self">nonnegative</a>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E3:102"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> holds <br/><font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1" target="_self">0.</a> 
 ;<br/>

<a NAME="E4:102"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> holds <br/><span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:102_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:102_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 ex <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>;<br/>
<a NAME="E3:102_1"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:102_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:102_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">s</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">s</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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:102_1_1_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:102_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">s'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E5:102_1_1_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">s'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">s'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T22" target="_self">Th22</a>, </i>;<br/>
<a NAME="E6:102_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E7:102_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E8:102_1_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">s'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1" target="_self">0.</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E9:102_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i><a class="ref" href="supinf_2.html#D6" target="_self">Def6</a>, <a class="ref" href="supinf_2.html#T23" target="_self">Th23</a>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:102_1_1"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:102_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:102"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:102_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:102_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:102_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:102_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T20" target="_self">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:102"/>
for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17" target="_self">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:102_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17" target="_self">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E2:102_3"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="supinf_2.html#K17" target="_self">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K6">ExtREAL</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:102_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">m</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">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:102_3"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
;<br/>
<b>hence </b><a NAME="E7:102_3"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:102"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17" target="_self">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<a NAME="E8:102"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n'</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17" target="_self">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E9:102"/><b>then </b>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" target="_self">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T80">SUPINF_1:80</a></i>;<br/>
<b>hence </b><a NAME="E10:102"/>
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 ;<br/>


</div>
