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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )</span><br/>









<b>assume </b><a NAME="E1:39"/>
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E2:39"/><b>then </b>
(  <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> &amp;  <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> &amp;  <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> &amp;  <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> )
 
<b>by </b><i><a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def4</a></i>;<br/>
<a NAME="E3:39"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="ref" href="mesfunc6.html#T99" title="MESFUNC6:th.99">MESFUNC6:99</a></i>;<br/>
<a NAME="E4:39"/><b>then </b><i><font color="Green" title="E26">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="ref" href="mesfun6c.html#T5" target="_self" title="MESFUN6C:th.5">COM19</a></i>;<br/>
<a NAME="E5:39"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/>
<a NAME="E6:39"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p3">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>
<a NAME="E7:39"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p3">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E4:39"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="measure1.html#T19" title="MEASURE1:th.19">MEASURE1:19</a></i>;<br/>
<b>hence </b><a NAME="E8:39"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="mesfun6c.html#T8" target="_self" title="MESFUN6C:th.8">COM99</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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