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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<a NAME="E3:29"/>
(  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp;  <a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
 
<b>by </b><i><a class="ref" href="integra4.html#T13" target="_self" title="INTEGRA4:th.13">Th13</a></i>;<br/>
<a NAME="E4:29"/><b>then </b><i><font color="Green" title="E30">A3</font></i>: 
(  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#T131" title="FUNCT_2:th.131">FUNCT_2:131</a></i>;<br/>
<a NAME="E5:29"/>
( <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="integra4.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E6:29"/><b>then </b>
(  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="integra4.html#T14" target="_self" title="INTEGRA4:th.14">Th14</a>, <a class="ref" href="integra4.html#T15" target="_self" title="INTEGRA4:th.15">Th15</a>, <a class="ref" href="integra4.html#T16" target="_self" title="INTEGRA4:th.16">Th16</a>, <a class="ref" href="integra4.html#T17" target="_self" title="INTEGRA4:th.17">Th17</a></i>;<br/>
<a NAME="E7:29"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
(  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E8:29"/><i><font color="Green" title="E32">A5</font></i>: 
(  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="integra4.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="integra4.html#E2:29"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="integra4.html#T20" target="_self" title="INTEGRA4:th.20">Th20</a>, <a class="ref" href="integra4.html#T22" target="_self" title="INTEGRA4:th.22">Th22</a></i>;<br/>
<a NAME="E9:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E4:29"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="integra4.html#E7:29"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="integra1.html#T59" title="INTEGRA1:th.59">INTEGRA1:59</a></i>;<br/>
<b>hence </b><a NAME="E10:29"/>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="rfunct_3.html#T37" title="RFUNCT_3:th.37">RFUNCT_3:37</a></i>;<br/>

<i><font color="Green" title="E33">A6</font></i>: <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> = 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E4:29"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="integra4.html#E7:29"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="integra4.html#E8:29"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="integra1.html#T59" title="INTEGRA1:th.59">INTEGRA1:59</a></i>
<br/>.= 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#T37" title="RFUNCT_3:th.37">RFUNCT_3:37</a></i>
;<br/>
<i><font color="Green" title="E34">A7</font></i>: <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> = 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E4:29"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="integra4.html#E7:29"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="integra4.html#E8:29"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="integra2.html#T33" title="INTEGRA2:th.33">INTEGRA2:33</a></i>
<br/>.= 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="ref" href="rfunct_3.html#T37" title="RFUNCT_3:th.37">RFUNCT_3:37</a></i>
;<br/>
<a NAME="E13:29"/><i><font color="Green" title="E35">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T40" title="RFUNCT_3:th.40">RFUNCT_3:40</a></i>;<br/>
<a NAME="E14:29"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T43" title="RFUNCT_3:th.43">RFUNCT_3:43</a></i>;<br/>
<a NAME="E15:29"/><b>then </b><i><font color="Green" title="E36">A9</font></i>: 
(  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="integra4.html#E4:29"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="integra4.html#E7:29"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="integra4.html#E8:29"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="integra4.html#E13:29"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="integra2.html#T32" title="INTEGRA2:th.32">INTEGRA2:32</a></i>;<br/>
<a NAME="E16:29"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E11:29"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="integra4.html#E12:29"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/>
<a NAME="E17:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="integra4.html#E15:29"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="real_2.html#T121" title="REAL_2:th.121">REAL_2:121</a></i>;<br/>
<a NAME="E18:29"/><b>then </b><i><font color="Green" title="E37">A10</font></i>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T105" title="REAL_2:th.105">REAL_2:105</a></i>;<br/>
<a NAME="E19:29"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max+</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E11:29"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="integra4.html#E12:29"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/>
<a NAME="E20:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="integra4.html#E15:29"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="real_2.html#T121" title="REAL_2:th.121">REAL_2:121</a></i>;<br/>
<a NAME="E21:29"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T110" title="REAL_2:th.110">REAL_2:110</a></i>;<br/>
<b>hence </b><a NAME="E22:29"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E18:29"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>


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