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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="firebrick">::  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/> ex <font color="Olive" title="b5">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font> st <br/>( <font color="Olive" title="b5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<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></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b5">E</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b2">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Olive" title="b3">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b5">E</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b2">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Olive" title="b4">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b5">E</font></span>)</span>)</span>)</span> )</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="firebrick">::  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/> ex <font color="Olive" title="b4">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<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></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">E</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b1">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">E</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b1">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Olive" title="b3">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">E</font></span>)</span>)</span>)</span> )</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="firebrick">::  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/> ex <font color="Olive" title="b3">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<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></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">E</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">E</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Olive" title="b2">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">E</font></span>)</span>)</span>)</span> )</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="firebrick">::  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  ex <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font></span>)</span>)</span>)</span> ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59"/><span class="lab"><font color="Green" title="E34">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="E2:59"/><span class="lab"><font color="Green" title="E35">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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font></span>)</span>)</span>)</span> )</span><br/>

<a NAME="E3:59"/>
( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="#E1:59"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="mesfun7c.html#T35" target="_self" title="MESFUN7C:th.35">Th35</a></span>;<br/>
<a NAME="E4:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="mesfunc6.html#T100" title="MESFUNC6:th.100">MESFUNC6:100</a></span>;<br/>
<a NAME="E5:59"/>
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">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="#E2:59"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="mesfun6c.html#D2" title="MESFUN6C:def.2">MESFUN6C:def 2</a></span>;<br/>
<a NAME="E6:59"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#D4" title="MESFUNC6:def.4">MESFUNC6:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">B2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E8:59"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
( <font color="Maroon" title="c6">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">B2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></span>;<br/>
<a NAME="E9:59"/>
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">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="#E1:59"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="mesfun6c.html#D2" title="MESFUN6C:def.2">MESFUN6C:def 2</a></span>;<br/>
<a NAME="E10:59"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#D4" title="MESFUNC6:def.4">MESFUNC6:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">A2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E12:59"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
( <font color="Maroon" title="c7">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c7">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></span>;<br/>
<a NAME="E13:59"/>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">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="#E2:59"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="mesfun6c.html#D2" title="MESFUN6C:def.2">MESFUN6C:def 2</a></span>;<br/>
<a NAME="E14:59"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#D4" title="MESFUNC6:def.4">MESFUNC6:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">B1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E16:59"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:59"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c8">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></span>;<br/>
<a NAME="E18:59"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:59"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">COMSEQ_3:def 3</a></span>;<br/>
<a NAME="E19:59"/>
<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> <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="#E1:59"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="mesfun6c.html#T33" title="MESFUN6C:th.33">MESFUN6C:33</a></span>;<br/>
<a NAME="E20:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="mesfun7c.html#T35" target="_self" title="MESFUN7C:th.35">Th35</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">G</font> = <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">F</font> = <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>;<br/>
<a NAME="E21:59"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun7c.html#T40" target="_self" title="MESFUN7C:th.40">Th40</a></span>;<br/>
<a NAME="E22:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun7c.html#T41" target="_self" title="MESFUN7C:th.41">Th41</a></span>;<br/>
<a NAME="E23:59"/>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">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="#E1:59"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="mesfun6c.html#D2" title="MESFUN6C:def.2">MESFUN6C:def 2</a></span>;<br/>
<a NAME="E24:59"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#D4" title="MESFUNC6:def.4">MESFUNC6:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">A1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E26:59"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E27:59"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c11">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></span>;<br/>
<a NAME="E28:59"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:59"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">COMSEQ_3:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">A</font> = <font color="Maroon" title="c11">A1</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c8">B1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E30:59"/>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c11">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:59"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="ref" href="mesfunc6.html#D1" title="MESFUNC6:def.1">MESFUNC6:def 1</a></span>;<br/>
<a NAME="E31:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A14</font></span>: 
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T16" title="MESFUNC6:th.16">MESFUNC6:16</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E32:59"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font></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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E33:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:59"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/>
<a NAME="E34:59"/>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c8">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:59"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="mesfunc6.html#D1" title="MESFUNC6:def.1">MESFUNC6:def 1</a></span>;<br/>
<a NAME="E35:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A17</font></span>: 
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T16" title="MESFUNC6:th.16">MESFUNC6:16</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E36:59"/>
( <font color="Maroon" title="c6">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">B2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a>, <a class="ref" href="mesfunc6.html#D1" title="MESFUNC6:def.1">MESFUNC6:def 1</a></span>;<br/>
<a NAME="E37:59"/><span class="kw">then </span>
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="mesfunc6.html#T16" title="MESFUNC6:th.16">MESFUNC6:16</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E38:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A18</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:59"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="ref" href="mesfun6c.html#D1" title="MESFUN6C:def.1">MESFUN6C:def 1</a></span>;<br/>
<a NAME="E39:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A19</font></span>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="mesfun6c.html#T30" title="MESFUN6C:th.30">MESFUN6C:30</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E40:59"/>
( <font color="Maroon" title="c7">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c7">A2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a>, <a class="ref" href="mesfunc6.html#D1" title="MESFUNC6:def.1">MESFUNC6:def 1</a></span>;<br/>
<a NAME="E41:59"/><span class="kw">then </span>
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:59"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="mesfunc6.html#T16" title="MESFUNC6:th.16">MESFUNC6:16</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E42:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A20</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:59"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="mesfun6c.html#D1" title="MESFUN6C:def.1">MESFUN6C:def 1</a></span>;<br/>
<a NAME="E43:59"/><span class="kw">then </span>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:59"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="mesfun6c.html#T30" title="MESFUN6C:th.30">MESFUN6C:30</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E44:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A21</font></span>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:59"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="mesfunc6.html#T26" title="MESFUNC6:th.26">MESFUNC6:26</a></span>;<br/>
<a NAME="E45:59"/>
<font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E46:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A22</font></span>: 
<font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:59"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/>
<a NAME="E47:59"/>
<font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E48:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A23</font></span>: 
<font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/>
<a NAME="E49:59"/><span class="lab"><font color="Green" title="E57">A24</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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#K1" title="RELSET_1:func.1">dom</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E50:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A25</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:59"><span class="lab"><font color="Green" title="E55">A22</font></span></a>, <a class="txt" href="#E48:59"><span class="lab"><font color="Green" title="E56">A23</font></span></a>, <a class="txt" href="#E33:59"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E51:59"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:59"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E42:59"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="txt" href="#E38:59"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="txt" href="#E32:59"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="mesfun6c.html#T11" title="MESFUN6C:th.11">MESFUN6C:11</a>, <a class="ref" href="mesfun6c.html#T30" title="MESFUN6C:th.30">MESFUN6C:30</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E53:59"/><span class="lab"><font color="Green" title="E59">A26</font></span>: 
<font color="Maroon" title="c13">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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#K1" title="RELSET_1:func.1">dom</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E54:59"/><span class="lab"><font color="Green" title="E60">A27</font></span>: 
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:59"><span class="lab"><font color="Green" title="E54">A21</font></span></a>, <a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E50:59"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="txt" href="#E20:59"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E22:59"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="mesfun6c.html#T42" title="MESFUN6C:th.42">MESFUN6C:42</a></span>;<br/>
<a NAME="E55:59"/><span class="lab"><font color="Green" title="E61">A28</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:59"><span class="lab"><font color="Green" title="E56">A23</font></span></a>, <a class="txt" href="#E50:59"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="txt" href="#E53:59"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">E</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>)</span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E56:59"/>
<font color="Maroon" title="c13">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:59"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E32:59"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E49:59"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="txt" href="#E46:59"><span class="lab"><font color="Green" title="E55">A22</font></span></a>, <a class="txt" href="#E48:59"><span class="lab"><font color="Green" title="E56">A23</font></span></a>, <a class="txt" href="#E33:59"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="txt" href="#E53:59"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>)</span>)</span></span><br/>

<a NAME="E57:59"/>
( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</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="#E1:59"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="mesfun7c.html#T35" target="_self" title="MESFUN7C:th.35">Th35</a>, <a class="ref" href="mesfunc6.html#T91" title="MESFUNC6:th.91">MESFUNC6:91</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">E1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E59:59"/><span class="lab"><font color="Green" title="E62">A29</font></span>: 
<font color="Maroon" title="c14">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E60:59"/><span class="lab"><font color="Green" title="E63">A30</font></span>: 
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">E1</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">E1</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T101" title="MESFUNC6:th.101">MESFUNC6:101</a></span>;<br/>
<a NAME="E61:59"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:59"><span class="lab"><font color="Green" title="E55">A22</font></span></a>, <a class="txt" href="#E50:59"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="txt" href="#E53:59"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E62:59"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">E1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">E1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:59"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="#E55:59"><span class="lab"><font color="Green" title="E61">A28</font></span></a>, <a class="ref" href="funct_1.html#T51" title="FUNCT_1:th.51">FUNCT_1:51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E63:59"/>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><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><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">E</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:59"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="txt" href="#E50:59"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="txt" href="#E53:59"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="txt" href="#E54:59"><span class="lab"><font color="Green" title="E60">A27</font></span></a>, <a class="txt" href="#E60:59"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="ref" href="mesfun7c.html#T39" target="_self" title="MESFUN7C:th.39">Th39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
