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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="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">r</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="E1:115"/>
( <font color="Maroon">f</font> <a href="mesfunc6.html#R3" target="_self">is_integrable_on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">f</font> <a href="mesfunc6.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E2:115"/><b>then </b>
(  <a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font> <a href="mesfunc5.html#R1">is_integrable_on</a> <font color="Maroon">M</font> &amp;  <a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:115"/><b>then </b>
( <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="mesfunc5.html#R1">is_integrable_on</a> <font color="Maroon">M</font> &amp;  <a href="mesfunc5.html#K8">Integral_on</a> <font color="Maroon">M</font>,<font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</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">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K8">Integral_on</a> <font color="Maroon">M</font>,<font color="Maroon">B</font>,<span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T118">MESFUNC5:118</a></i>;<br/>
<a NAME="E4:115"/><b>then </b>
(  <a href="mesfunc5.html#K1">R_EAL</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> <a href="mesfunc5.html#R1">is_integrable_on</a> <font color="Maroon">M</font> &amp;  <a href="mesfunc5.html#K8">Integral_on</a> <font color="Maroon">M</font>,<font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</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">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K8">Integral_on</a> <font color="Maroon">M</font>,<font color="Maroon">B</font>,<span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:115"/>
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="mesfunc6.html#R3" target="_self">is_integrable_on</a> <font color="Maroon">M</font> &amp;  <a href="mesfunc6.html#K7" target="_self">Integral_on</a> <font color="Maroon">M</font>,<font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</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">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K7" 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/>;<br/>


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