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

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

<b>assume </b><a NAME="E1:27"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">eps</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x0</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E18</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">eps</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green">E19</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E5:27"><i><font color="Green">E19</font></i></a>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</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>for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="supinf_1.html#NM3">R_eal</a><br/> for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K3" target="_self">len</a> <font color="Olive">a</font>;<br/>
<a NAME="E6:27"/><i><font color="Green">E22</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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>1</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="27_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/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>take </b>
 <a href="measure6.html#K3" target="_self">len</a> <font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E1:27_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>, <a href="measure6.html#K3" target="_self">len</a> <font color="Maroon">x</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E8:27"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &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>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">n</font>,<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</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#S2">RECDEF_1:sch 2</a>();<br/></i>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E9:27"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i>, <a class="txt" href="measure6.html#E6:27"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E10:27"/><i><font color="Green">E33</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>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</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="27_2"><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="E1:27_2"/><i><font color="Green">E34</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:27_2"/>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</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>  <a href="measure6.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure6.html#E6:27"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:27_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="measure6.html#E8:27"><i><font color="Green">E25</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E11:27"/><i><font color="Green">E35</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">S<sub>2</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>

<a NAME="E12:27"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">n</font>
 
;<br/>
<a NAME="E13:27"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E14:27"/>
for <font color="Olive">s</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:27_3"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_3"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</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">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="measure6.html#T8" target="_self">Th8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font>;<br/>
<a NAME="E5:27_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i>, <a class="txt" href="measure6.html#E6:27"><i><font color="Green">E22</font></i></a>, <a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<a NAME="E6:27_3"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">l</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">l</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:27_3_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E2:27_3_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<a NAME="E3:27_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure6.html#E6:27"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E4:27_3_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E8:27"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E5:27_3_1"/><b>then </b><i><font color="Green">E72</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font> )
 
<b>by </b><i>, <a class="txt" href="measure6.html#E3:27"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E6:27_3_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="measure6.html#E8:27"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E7:27_3_1"/><b>then </b>
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure1.html#T4">MEASURE1:4</a></i>;<br/>
<a NAME="E8:27_3_1"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<a NAME="E9:27_3_1"/><i><font color="Green">E74</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E9:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="supinf_2.html#T59">SUPINF_2:59</a></i>;<br/>
<a NAME="E10:27_3_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure6.html#E10:27"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="measure1.html#T4">MEASURE1:4</a></i>;<br/>
<a NAME="E11:27_3_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="measure6.html#E8:27"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E12:27_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure6.html#E11:27"><i><font color="Green">E35</font></i></a>, <a class="ref" href="measure6.html#T11" target="_self">Th11</a>, <a class="ref" href="measure4.html#T1">MEASURE4:1</a></i>;<br/>
<b>hence </b><a NAME="E13:27_3_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="measure6.html#T10" target="_self">Th10</a>, , <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:27_3"/>
for <font color="Olive">k</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">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E8:27_3"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font>
 
;<br/>
<a NAME="E9:27_3"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E8:27"><i><font color="Green">E25</font></i></a>, <a class="txt" href="measure6.html#E9:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="supinf_2.html#T59">SUPINF_2:59</a></i>;<br/>
<b>hence </b><a NAME="E10:27_3"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="measure6.html#T8" target="_self">Th8</a>, <a class="ref" href="measure6.html#T10" target="_self">Th10</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:27"/><b>then </b>
<font color="Maroon">x0</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">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="E16:27"/><b>then </b><i><font color="Green">E77</font></i>: 
(  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> &amp;  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a>, <a class="ref" href="supinf_2.html#D23">SUPINF_2:def 23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_4"/>
(  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> or  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> )
 </span><b>by </b><i><a class="ref" href="measure6.html#T8" target="_self">Th8</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_1"><b>suppose </b></a><a NAME="E1:27_4_1"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_4_1"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">eps</font>
 <b>by </b><i><a class="txt" href="measure6.html#E5:27"><i><font color="Green">E19</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_2"><b>suppose </b></a><a NAME="E1:27_4_2"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_4_2"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">eps</font>
 <b>by </b><i><a class="txt" href="measure6.html#E5:27"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
