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











<b>assume </b><a NAME="E1:151"/><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> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E2:151"/><i><font color="Green">E37</font></i>: 
 ex <font color="Olive">A</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">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="Olive">A</font> )
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">BF</font> = <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">B</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a></i>;<br/>
<a NAME="E4:151"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:151"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</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="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:151"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</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">B</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E7:151"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <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="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E8:151"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E9:151"/>
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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_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="E1:151_1"/><i><font color="Green">E40</font></i>: 
<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"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<a NAME="E2:151_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</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="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:151_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</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">B</font> &amp; <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"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E4:151_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<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"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E5:151_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<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"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E6:151_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:151_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <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"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E8:151_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E9:151_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:151"/><b>then </b>
<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="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E11:151"/>
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font> &amp;  <a href="mesfunc5.html#K8" target="_self">Integral_on</a> <font color="Maroon">M</font>,<font color="Maroon">B</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> <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#K8" target="_self">Integral_on</a> <font color="Maroon">M</font>,<font color="Maroon">B</font>,<font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div>
