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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>;<br/>









<b>assume </b><a NAME="E1:148"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font>
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:148"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E4:148"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a>, , </i>;<br/>
<a NAME="E5:148"/><i><font color="Green">E39</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">I1</font> =  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">I2</font> =  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<a NAME="E8:148"/><i><font color="Green">E40</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp;  <a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:148"/><i><font color="Green">E41</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a>, </i>;<br/>
<a NAME="E10:148"/><i><font color="Green">E42</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> &amp;  <a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="mesfunc2.html#T27">MESFUNC2:27</a>, <a class="ref" href="mesfunc2.html#T28">MESFUNC2:28</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:148_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> or <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1"><b>suppose </b></a><a NAME="E1:148_1_1"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><a NAME="E2:148_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:148_1_1"/><i><font color="Green">E47</font></i>: 
(  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E4:148_1_1"/>
( <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <font color="Maroon">I1</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <font color="Maroon">I2</font> is   <a href="real_1.html#NM1">Real</a> )
 ;<br/><a NAME="E5:148_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> is   <a href="real_1.html#NM1">Real</a> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><a NAME="E6:148_1_1"/><b>then </b>
(  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, , <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><b>hence </b><a NAME="E7:148_1_1"/>
<font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font>
 <b>by </b><i>, , , <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/><b>thus </b> <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="extreal1.html#T30">EXTREAL1:30</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_2"><b>suppose </b></a><a NAME="E1:148_1_2"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:148_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="real_1.html#K1">-</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:148_1_2"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:148_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E6:148_1_2"/><i><font color="Green">E98</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E3:148"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E7:148_1_2"/><i><font color="Green">E99</font></i>: 
(  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E8:148_1_2"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">I2</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">I1</font> is   <a href="real_1.html#NM1">Real</a> )
 ;<br/><a NAME="E9:148_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> is   <a href="real_1.html#NM1">Real</a> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><a NAME="E10:148_1_2"/><b>then </b>
(  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, , <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><b>hence </b><a NAME="E11:148_1_2"/>
<font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font>
 <b>by </b><i>, , , <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E12:148_1_2"/><i><font color="Green">E100</font></i>: 
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T3">SUPINF_2:3</a></i>;<br/><b>thus </b> <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="extreal1.html#T30">EXTREAL1:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="extreal2.html#T15">EXTREAL2:15</a></i>
<br/>.= 
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="extreal1.html#T26">EXTREAL1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="extreal1.html#T26">EXTREAL1:26</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
