<?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="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:29"/>
(  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:29"/><b>then </b><i><font color="Green">E40</font></i>: 
(  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a>  &amp;  <a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</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="E5:29"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</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#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E6:29"/><b>then </b>
(  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp;  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> &amp;  <a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp;  <a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E7:29"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> &amp;  <a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</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/>
<a NAME="E8:29"/><i><font color="Green">E42</font></i>: 
(  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp;  <a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E9:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</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="integra1.html#T59">INTEGRA1:59</a></i>;<br/>
<b>hence </b><a NAME="E10:29"/>
 <a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rfunct_3.html#T37">RFUNCT_3:37</a></i>;<br/>

<i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="integra1.html#T59">INTEGRA1:59</a></i>
<br/>.= 
 <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#T37">RFUNCT_3:37</a></i>
;<br/>
<i><font color="Green">E44</font></i>: <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="integra2.html#T33">INTEGRA2:33</a></i>
<br/>.= 
 <a href="integra1.html#K13">integral</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="rfunct_3.html#T37">RFUNCT_3:37</a></i>
;<br/>
<a NAME="E13:29"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</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> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:29_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>thus </b><a NAME="E2:29_3"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</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> 0
 <b>by </b><i><a class="ref" href="integra4.html#T1" target="_self">Th1</a>, <a class="ref" href="rfunct_3.html#T40">RFUNCT_3:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:29"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</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> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:29_4"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>thus </b><a NAME="E2:29_4"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</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> 0
 <b>by </b><i><a class="ref" href="integra4.html#T1" target="_self">Th1</a>, <a class="ref" href="rfunct_3.html#T43">RFUNCT_3:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:29"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, , , , <a class="ref" href="integra2.html#T32">INTEGRA2:32</a></i>;<br/>
<a NAME="E16:29"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra4.html#T1" target="_self">Th1</a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<a NAME="E18:29"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K13">integral</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<a NAME="E19:29"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra4.html#T1" target="_self">Th1</a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<a NAME="E21:29"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K13">integral</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T110">REAL_2:110</a></i>;<br/>
<b>hence </b><a NAME="E22:29"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>


</div>
