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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Olive" title="b3">f</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b4">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Olive" title="b2">M</font> holds <br/><font color="Olive" title="b3">f</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Olive" title="b4">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Olive" title="b2">f</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Olive" title="b1">M</font> holds <br/><font color="Olive" title="b2">f</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Olive" title="b3">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Olive" title="b1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Olive" title="b1">f</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b2">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> holds <br/><font color="Olive" title="b1">f</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Olive" title="b2">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c4">f</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c5">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E2:42"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c5">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E3:42"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
 <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def4</a></span>;<br/>
<a NAME="E4:42"/>
 <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def4</a></span>;<br/>
<a NAME="E5:42"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="mesfun6c.html#T34" target="_self" title="MESFUN6C:th.34">Th34</a></span>;<br/>
<a NAME="E6:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A4</font></span>: 
 <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T6" target="_self" title="MESFUN6C:th.6">Th6</a></span>;<br/>
<a NAME="E7:42"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def4</a></span>;<br/>
<a NAME="E8:42"/>
 <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def4</a></span>;<br/>
<a NAME="E9:42"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="mesfun6c.html#T34" target="_self" title="MESFUN6C:th.34">Th34</a></span>;<br/>
<a NAME="E10:42"/><span class="kw">then </span>
 <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T6" target="_self" title="MESFUN6C:th.6">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:42"/>
<font color="Maroon" title="c4">f</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c5">g</font> <a href="mesfun6c.html#R2" title="MESFUN6C:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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