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

<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#R5">is_monotone_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1"/>
( <font color="Maroon">f</font> <a href="rfunct_2.html#R3">is_non_decreasing_on</a> <font color="Maroon">A</font> or <font color="Maroon">f</font> <a href="rfunct_2.html#R4">is_non_increasing_on</a> <font color="Maroon">A</font> )
 </span><b>by </b><i>, <a class="ref" href="rfunct_2.html#D6">RFUNCT_2:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>suppose </b></a><a NAME="E1:20_1_1"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#R3">is_non_decreasing_on</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1"/>
<font color="Maroon">f</font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2"><b>suppose </b></a><a NAME="E1:20_1_2"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#R4">is_non_increasing_on</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:20_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="rfunct_2.html#R3">is_non_decreasing_on</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T67">RFUNCT_2:67</a></i>;<br/><a NAME="E3:20_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="seq_1.html#K16">-</a> <font color="Maroon">f</font> <a href="rfunct_2.html#R3">is_non_decreasing_on</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T38">RFUNCT_1:38</a></i>;<br/><a NAME="E4:20_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>;<br/><a NAME="E5:20_1_2"/><b>then </b>
 <a href="seq_1.html#K16">-</a> <font color="Maroon">f</font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:20_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E7:20_1_2"/><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>proof </b></a><div class="add">

<i><font color="Green">E52</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E2:20_1_2_1"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<b>consider </b><font color="Maroon">x</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:20_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:20_1_2_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E6:20_1_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:20_1_2_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_2_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:20_1_2_1_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<a NAME="E3:20_1_2_1_2_1"/><i><font color="Green">E76</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<font color="Maroon">A</font> = 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
;<br/>
<a NAME="E5:20_1_2_1_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E7:20_1_2_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E8:20_1_2_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:20_1_2_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E10:20_1_2_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E11:20_1_2_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="rfunct_2.html#T48">RFUNCT_2:48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D9">RFUNCT_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:20_1_2_1"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_2_1_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_2_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:20_1_2_1_3_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<a NAME="E3:20_1_2_1_3_1"/><i><font color="Green">E81</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<font color="Maroon">A</font> = 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
;<br/>
<a NAME="E5:20_1_2_1_3_1"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E7:20_1_2_1_3_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E8:20_1_2_1_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:20_1_2_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E10:20_1_2_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E11:20_1_2_1_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="rfunct_2.html#T48">RFUNCT_2:48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_2_1_3"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D10">RFUNCT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:20_1_2_1"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:20_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/><a NAME="E9:20_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="integra2.html#T31">INTEGRA2:31</a></i>;<br/><a NAME="E10:20_1_2"/>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_2_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> &amp; ( for <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_2_1"><b>proof </b></a><div class="add">

<i><font color="Green">E85</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E2:20_1_2_2_1"/><i><font color="Green">E86</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>

<a NAME="E3:20_1_2_2_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E4:20_1_2_2_1"/><i><font color="Green">E87</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>

<a NAME="E5:20_1_2_2_1"/>
for <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:20_1_2_2_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:20_1_2_2_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:20_1_2_2_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 
;<br/>
<a NAME="E4:20_1_2_2_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
;<br/>
<b>then </b><i><font color="Green">E92</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font>

;<br/>
<a NAME="E7:20_1_2_2_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:20_1_2_2_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 
;<br/>
<a NAME="E9:20_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E10:20_1_2_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:20_1_2_2_1"/>
for <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_2_2"/>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:20_1_2"/>
<font color="Maroon">f</font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
