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

<b>let </b><font color="Maroon">seq</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">Eseq</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="E1:17"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">seq</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">Eseq</font> &amp; <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 ;<br/>

<a NAME="E2:17"/><i><font color="Green">E36</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Eseq</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">seq</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/>
<b>reconsider </b><font color="Maroon">s</font> =  <a href="rinfsup1.html#K2">inf</a> <font color="Maroon">seq</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E4:17"/>
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font> holds <br/><font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">Eseq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</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">n1</font> = <font color="Maroon">n1</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="E5:17_1"/>
 <a href="rinfsup1.html#K2">inf</a> <font color="Maroon">seq</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font>
 
<b>by </b><i>, <a class="ref" href="rinfsup1.html#T8">RINFSUP1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">s</font> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E6:17"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font> is <a href="supinf_1.html#V4">bounded_below</a>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D12">SUPINF_1:def 12</a></i>;<br/>
<a NAME="E7:17"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">k1</font> =  <a href="supinf_1.html#K1">+infty</a>  as    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font> <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">n1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:17_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">Eseq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</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">n1</font> = <font color="Maroon">n1</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="E6:17_2"/>
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:17_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:17"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T84">SUPINF_1:84</a></i>;<br/>
<a NAME="E9:17"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Eseq1</font> = <font color="Maroon">Eseq</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K6">ExtREAL</a>  ;<br/>
<a NAME="E11:17"/>
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<a NAME="E1:17_3"/><i><font color="Green">E55</font></i>: 
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font></span>)</span> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> =  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, , <a class="ref" href="supinf_1.html#T84">SUPINF_1:84</a></i>;<br/>
<a NAME="E3:17_3"/>
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">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_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="E1:17_3_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Eseq</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:17_3_1"/>
<font color="Maroon">Eseq</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:17_3_1"/><b>then </b>
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Eseq</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:17_3_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font></span>)</span> &amp; <font color="Maroon">r3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Eseq</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r3</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E6:17_3_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:17_3"/><b>then </b>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K2">inf</a> <font color="Maroon">seq</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T10">RINFSUP1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:17_3"/>
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:17"/>
 <a href="rinfsup1.html#K2">inf</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Eseq</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div>
