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

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</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><a NAME="E1:28"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 ;<br/>

<a NAME="E2:28"/><i><font color="Green">E34</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="integra6.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E3:28"/><i><font color="Green">E35</font></i>: 
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K2">integral</a> <font color="Maroon">f</font>,<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i><a class="ref" href="integra6.html#T1" target="_self">Th1</a>, <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>suppose </b></a><a NAME="E1:28_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">d</font>,<font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="integra6.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><a NAME="E1:28_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 ;<br/><div class="add"><a NAME="E2:28_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="integra6.html#T1" target="_self">Th1</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:28_1_2"/><b>then </b>
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">d</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2">integral</a> <font color="Maroon">f</font>,<span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="integra5.html#K3">']</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>;<br/><b>hence </b><a NAME="E4:28_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">d</font>,<font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="integra6.html#E1:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="integra6.html#T2" target="_self">Th2</a>, <a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
