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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> ;<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>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>;<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="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c6">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>;<br/>











<b>assume </b><a NAME="E1:113"/><i><font color="Green" title="E33">A1</font></i>: 
( <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; <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; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 ;<br/>

<a NAME="E2:113"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
(  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="ref" href="mesfunc6.html#D9" target="_self" title="MESFUNC6:def.9">Def9</a></i>;<br/>
 <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="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>

<br/>.= 
 <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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc6.html#T23" target="_self" title="MESFUNC6:th.23">Th23</a></i>
;<br/>
<a NAME="E4:113"/><b>then </b>
( <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc5.html#K8" title="MESFUNC5:func.8">Integral_on</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K8" title="MESFUNC5:func.8">Integral_on</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">B</font>,<span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K8" title="MESFUNC5:func.8">Integral_on</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">B</font>,<span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc6.html#E1:113"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="mesfunc6.html#E2:113"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="mesfunc5.html#T117" title="MESFUNC5:th.117">MESFUNC5:117</a></i>;<br/>
<a NAME="E5:113"/><b>then </b>
(  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc5.html#K8" title="MESFUNC5:func.8">Integral_on</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">B</font>,<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K8" title="MESFUNC5:func.8">Integral_on</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">B</font>,<span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K8" title="MESFUNC5:func.8">Integral_on</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">B</font>,<span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc6.html#T23" target="_self" title="MESFUNC6:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E6:113"/>
( <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</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="mesfunc6.html#K7" title="MESFUNC6:func.7">Integral_on</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K7" title="MESFUNC6:func.7">Integral_on</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K7" title="MESFUNC6:func.7">Integral_on</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c5">g</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc6.html#D9" target="_self" title="MESFUNC6:def.9">Def9</a></i>;<br/>


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