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<span class="kw">let </span><font color="Maroon" title="c2">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c2">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="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b4">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b2">M</font> holds <br/> <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b2">M</font>,<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="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="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"><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>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</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"><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>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">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="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b2">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b1">M</font> holds <br/> <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b1">M</font>,<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="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="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"><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>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</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"><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>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> holds <br/> <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<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="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="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="c4">M</font>,<span class="p2"><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>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2"><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>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> implies  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<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="c5">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="c4">M</font>,<span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>)</span>)</span> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:84"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E2:84"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c4">M</font>,<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="c5">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="c4">M</font>,<span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>)</span>)</span></span><br/>

<a NAME="E3:84"/>
 ex <font color="Olive" title="b1">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> st <br/>( <font color="Maroon" title="c5">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">f1</font> &amp;  ex <font color="Olive" title="b2">NDf</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">NDf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">NDf</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b1">f1</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E52">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">NDf1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E5:84"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">NDf1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:84"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">NDf1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> )
 ;<br/>
<a NAME="E7:84"/>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="lpspacc1.html#T37" target="_self" onmouseover="rs('lpspacc1/T37')" onmouseout="rh()">Th37</a>, <a class="ref" href="mesfunc6.html#D4" onmouseover="rs('mesfunc6/D4')" onmouseout="rh()">MESFUNC6:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Ef</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E9:84"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c8">Ef</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"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:84"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <font color="Maroon" title="c8">Ef</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 ;<br/>
<a NAME="E11:84"/>
 ex <font color="Olive" title="b1">g1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> st <br/>( <font color="Maroon" title="c6">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">g1</font> &amp;  ex <font color="Olive" title="b2">NDg</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">NDg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">NDg</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b1">g1</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E53">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">NDg1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E13:84"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">NDg1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E14:84"/><span class="lab"><font color="Green" title="E59">A8</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">NDg1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">g</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> )
 ;<br/>
<a NAME="E15:84"/>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:84"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="lpspacc1.html#T37" target="_self" onmouseover="rs('lpspacc1/T37')" onmouseout="rh()">Th37</a>, <a class="ref" href="mesfunc6.html#D4" onmouseover="rs('mesfunc6/D4')" onmouseout="rh()">MESFUNC6:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">Eg</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E17:84"/><span class="lab"><font color="Green" title="E60">A9</font></span>: 
<font color="Maroon" title="c10">Eg</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"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E18:84"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <font color="Maroon" title="c10">Eg</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E20:84"/><span class="lab"><font color="Green" title="E62">A11</font></span>: 
( <font color="Maroon" title="c11">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="c5">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c6">g</font></span>)</span> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">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="c5">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c6">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="c11">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="c4">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">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="c11">E</font></span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">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="c6">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="c11">E</font></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E14:84"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="mesfun7c.html#T42" onmouseover="rs('mesfun7c/T42')" onmouseout="rh()">MESFUN7C:42</a></span>;<br/>
<a NAME="E21:84"/><span class="lab"><font color="Green" title="E63">A12</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="c5">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c6">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="c11">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:84"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="ref" href="valued_1.html#D11" onmouseover="rs('valued_1/D11')" onmouseout="rh()">VALUED_1:def 11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">NF</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">NDf1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">NDg1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">NG</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">NDg1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">NDf1</font>;<br/>
<a NAME="E22:84"/>
( <font color="Maroon" title="c7">NDf1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c9">NDg1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#D1" onmouseover="rs('measure1/D1')" onmouseout="rh()">MEASURE1:def 1</a></span>;<br/>
<a NAME="E23:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A13</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">NDf1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">NDg1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">NDg1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">NDf1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c8">Ef</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c10">Eg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T11" onmouseover="rs('measure1/T11')" onmouseout="rh()">MEASURE1:11</a></span>;<br/>
<a NAME="E24:84"/><span class="lab"><font color="Green" title="E65">A14</font></span>: 
<font color="Maroon" title="c8">Ef</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">NDf1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E9:84"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="valued_1.html#D11" onmouseover="rs('valued_1/D11')" onmouseout="rh()">VALUED_1:def 11</a></span>;<br/>
<a NAME="E25:84"/><span class="lab"><font color="Green" title="E66">A15</font></span>: 
<font color="Maroon" title="c10">Eg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">NDg1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:84"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E17:84"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="ref" href="valued_1.html#D11" onmouseover="rs('valued_1/D11')" onmouseout="rh()">VALUED_1:def 11</a></span>;<br/>
<a NAME="E26:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A16</font></span>: 
<font color="Maroon" title="c11">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c10">Eg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:84"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E24:84"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E14:84"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E68">A17</font></span>: <font color="Maroon" title="c8">Ef</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c10">Eg</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">NDf1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">NDf1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">NDg1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:84"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E25:84"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="ref" href="xboole_1.html#T52" onmouseover="rs('xboole_1/T52')" onmouseout="rh()">XBOOLE_1:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">NDf1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">NDf1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">NDg1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T41" onmouseover="rs('xboole_1/T41')" onmouseout="rh()">XBOOLE_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">NDf1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">NDg1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">NDf1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">NDg1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" onmouseover="rs('xboole_1/T37')" onmouseout="rh()">XBOOLE_1:37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">NDf1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">NDg1</font>

;<br/>
<a NAME="E28:84"/><span class="lab"><font color="Green" title="E69">A18</font></span>: 
<font color="Maroon" title="c11">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Ef</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Ef</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c10">Eg</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:84"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="ref" href="xboole_1.html#T48" onmouseover="rs('xboole_1/T48')" onmouseout="rh()">XBOOLE_1:48</a></span>;<br/>
<span class="lab"><font color="Green" title="E70">A19</font></span>: <font color="Maroon" title="c10">Eg</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c8">Ef</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">NDg1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">NDg1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">NDf1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:84"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E25:84"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="ref" href="xboole_1.html#T52" onmouseover="rs('xboole_1/T52')" onmouseout="rh()">XBOOLE_1:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">NDg1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">NDg1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">NDf1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T41" onmouseover="rs('xboole_1/T41')" onmouseout="rh()">XBOOLE_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">NDg1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">NDf1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">NDg1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">NDf1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" onmouseover="rs('xboole_1/T37')" onmouseout="rh()">XBOOLE_1:37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">NDg1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">NDf1</font>

;<br/>
<a NAME="E30:84"/><span class="lab"><font color="Green" title="E71">A20</font></span>: 
<font color="Maroon" title="c11">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Eg</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Eg</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c8">Ef</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:84"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="ref" href="xboole_1.html#T48" onmouseover="rs('xboole_1/T48')" onmouseout="rh()">XBOOLE_1:48</a></span>;<br/>
<a NAME="E31:84"/>
( <font color="Maroon" title="c7">NDf1</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c9">NDg1</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c4">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E13:84"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="ref" href="measure1.html#D7" onmouseover="rs('measure1/D7')" onmouseout="rh()">MEASURE1:def 7</a></span>;<br/>
<a NAME="E32:84"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">NDf1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">NDg1</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c4">M</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">NDg1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">NDf1</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c4">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:84"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="ref" href="measure1.html#T36" onmouseover="rs('measure1/T36')" onmouseout="rh()">MEASURE1:36</a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E33:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A21</font></span>: 
( <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">NDf1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">NDg1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">NDg1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">NDf1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#D7" onmouseover="rs('measure1/D7')" onmouseout="rh()">MEASURE1:def 7</a></span>;<br/>
<a NAME="E34:84"/><span class="lab"><font color="Green" title="E73">A22</font></span>: 
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">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"><font color="Maroon" title="c5">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="c11">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">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="#E28:84"><span class="lab"><font color="Green" title="E69">A18</font></span></a>, <a class="txt" href="#E27:84"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="txt" href="#E10:84"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a>, <a class="txt" href="#E9:84"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E33:84"><span class="lab"><font color="Green" title="E72">A21</font></span></a>, <a class="ref" href="mesfunc6.html#T89" onmouseover="rs('mesfunc6/T89')" onmouseout="rh()">MESFUNC6:89</a></span>;<br/>
<a NAME="E35:84"/>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">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"><font color="Maroon" title="c6">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="c11">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c6">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="#E18:84"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a>, <a class="txt" href="#E17:84"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E33:84"><span class="lab"><font color="Green" title="E72">A21</font></span></a>, <a class="txt" href="#E30:84"><span class="lab"><font color="Green" title="E71">A20</font></span></a>, <a class="txt" href="#E29:84"><span class="lab"><font color="Green" title="E70">A19</font></span></a>, <a class="ref" href="mesfunc6.html#T89" onmouseover="rs('mesfunc6/T89')" onmouseout="rh()">MESFUNC6:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:84"/>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<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="c5">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="c4">M</font>,<span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c6">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="txt" href="#E20:84"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E21:84"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="relat_1.html#T69" onmouseover="rs('relat_1/T69')" onmouseout="rh()">RELAT_1:69</a>, <a class="txt" href="#E34:84"><span class="lab"><font color="Green" title="E73">A22</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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