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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#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="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#K1" title="NUMBERS:func.1">REAL</a><br/>  for <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 <font color="Olive" title="b2">M</font> is  <a href="measure3.html#V1" title="MEASURE3:attr.1">complete</a>  &amp; <font color="Olive" title="b3">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b3">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">M</font> &amp; <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> <font color="Olive" title="b3">f</font> &amp; <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> <font color="Olive" title="b4">g</font> holds <br/>( <font color="Olive" title="b4">g</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b2">M</font> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b4">g</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" 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="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#K1" title="NUMBERS:func.1">REAL</a><br/>  for <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 <font color="Olive" title="b1">M</font> is  <a href="measure3.html#V1" title="MEASURE3:attr.1">complete</a>  &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b2">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">M</font> &amp; <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> <font color="Olive" title="b2">f</font> &amp; <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> <font color="Olive" title="b3">g</font> holds <br/>( <font color="Olive" title="b3">g</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b1">M</font> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b3">g</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" 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="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <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 <font color="Maroon" title="c3">M</font> is  <a href="measure3.html#V1" title="MEASURE3:attr.1">complete</a>  &amp; <font color="Olive" title="b1">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b1">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c3">M</font> &amp; <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> <font color="Olive" title="b1">f</font> &amp; <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> <font color="Olive" title="b2">g</font> holds <br/>( <font color="Olive" title="b2">g</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">f</font>) <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="c3">M</font>,<font color="Olive" title="b2">g</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">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 <font color="Maroon" title="c3">M</font> is  <a href="measure3.html#V1" title="MEASURE3:attr.1">complete</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> &amp; <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> <font color="Maroon" title="c4">f</font> &amp; <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> <font color="Maroon" title="c5">g</font> holds <br/>( <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <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="c3">M</font>,<font color="Maroon" title="c5">g</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c3">M</font> is  <a href="measure3.html#V1" title="MEASURE3:attr.1">complete</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c6">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> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">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> <font color="Maroon" title="c5">g</font> implies ( <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <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="c3">M</font>,<font color="Maroon" title="c5">g</font>) ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c3">M</font> is  <a href="measure3.html#V1" title="MEASURE3:attr.1">complete</a> 
 <span class="kw">and </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E4:50"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c6">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> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E5:50"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c6">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> <font color="Maroon" title="c5">g</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"> ( <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <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="c3">M</font>,<font color="Maroon" title="c5">g</font>) )</span><br/>

<a NAME="E6:50"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E45">A2</font></span></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="c7">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="E8:50"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
( <font color="Maroon" title="c7">E1</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> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font> is <font color="Maroon" title="c7">E1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D17" onmouseover="rs('mesfunc5/D17')" onmouseout="rh()">MESFUNC5:def 17</a></span>;<br/>
<a NAME="E9:50"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="mesfunc5.html#D17" onmouseover="rs('mesfunc5/D17')" onmouseout="rh()">MESFUNC5:def 17</a></span>;<br/>
<a NAME="E10:50"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="mesfunc5.html#D17" onmouseover="rs('mesfunc5/D17')" onmouseout="rh()">MESFUNC5:def 17</a></span>;<br/>
<a NAME="E11:50"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
(  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D7" onmouseover="rs('mesfunc5/D7')" onmouseout="rh()">MESFUNC5:def 7</a></span>;<br/>
<a NAME="E12:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A11</font></span>: 
<font color="Maroon" title="c4">f</font> is <font color="Maroon" title="c6">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a></span>;<br/>
<a NAME="E13:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A12</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font> is <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="mesfun14.html#T26" target="_self" onmouseover="rs('mesfun14/T26')" onmouseout="rh()">Th26</a>, <a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a></span>;<br/>
<a NAME="E14:50"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
( <font color="Maroon" title="c6">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"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c6">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"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="rfunct_3.html#D10" onmouseover="rs('rfunct_3/D10')" onmouseout="rh()">RFUNCT_3:def 10</a></span>;<br/>
<a NAME="E15:50"/>
(  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> is <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp;  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font></span>)</span> is <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="txt" href="#E13:50"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="mesfun11.html#T10" onmouseover="rs('mesfun11/T10')" onmouseout="rh()">MESFUN11:10</a></span>;<br/>
<a NAME="E16:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A14</font></span>: 
(  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> is <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp;  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> is <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun14.html#T43" target="_self" onmouseover="rs('mesfun14/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E17:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A15</font></span>: 
(  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font> is <font color="Maroon" title="c6">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  &amp;  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font> is <font color="Maroon" title="c6">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a></span>;<br/>
<a NAME="E18:50"/><span class="kw">then </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"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="ref" href="mesfunc6.html#T61" onmouseover="rs('mesfunc6/T61')" onmouseout="rh()">MESFUNC6:61</a>, <a class="ref" href="mesfunc6.html#T82" onmouseover="rs('mesfunc6/T82')" onmouseout="rh()">MESFUNC6:82</a></span>;<br/>
<a NAME="E19:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A16</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"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="mesfun14.html#T43" target="_self" onmouseover="rs('mesfun14/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">E0</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="E21:50"/><span class="lab"><font color="Green" title="E60">A17</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">E0</font> <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">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">E2</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="E23:50"/><span class="lab"><font color="Green" title="E61">A18</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">E2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">E2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="mesfun14.html#T42" target="_self" onmouseover="rs('mesfun14/T42')" onmouseout="rh()">Th42</a>, <a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</a></span>;<br/>
<a NAME="E24:50"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">E2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">E2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<a NAME="E25:50"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">E2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">E2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:50"><span class="lab"><font color="Green" title="E61">A18</font></span></a>, <a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<a NAME="E26:50"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c9">E2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">E2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c9">E2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E27:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A19</font></span>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c9">E2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c9">E2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E28:50"/>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</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="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c9">E2</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E16:50"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E23:50"><span class="lab"><font color="Green" title="E61">A18</font></span></a>, <a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a>, <a class="ref" href="mesfunc6.html#T89" onmouseover="rs('mesfunc6/T89')" onmouseout="rh()">MESFUNC6:89</a></span>;<br/>
<a NAME="E29:50"/><span class="kw">then </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"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</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="c3">M</font>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E16:50"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E23:50"><span class="lab"><font color="Green" title="E61">A18</font></span></a>, <a class="txt" href="#E27:50"><span class="lab"><font color="Green" title="E62">A19</font></span></a>, <a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a>, <a class="ref" href="mesfunc6.html#T89" onmouseover="rs('mesfunc6/T89')" onmouseout="rh()">MESFUNC6:89</a></span>;<br/>
<a NAME="E30:50"/><span class="kw">then </span>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E17:50"><span class="lab"><font color="Green" title="E58">A15</font></span></a>, <a class="txt" href="#E19:50"><span class="lab"><font color="Green" title="E59">A16</font></span></a>, <a class="ref" href="mesfunc6.html#T61" onmouseover="rs('mesfunc6/T61')" onmouseout="rh()">MESFUNC6:61</a>, <a class="ref" href="mesfunc6.html#T82" onmouseover="rs('mesfunc6/T82')" onmouseout="rh()">MESFUNC6:82</a></span>;<br/>
<a NAME="E31:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A20</font></span>: 
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun14.html#T43" target="_self" onmouseover="rs('mesfun14/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E32:50"/><span class="lab"><font color="Green" title="E64">A21</font></span>: 
( <font color="Maroon" title="c6">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"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c6">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"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="rfunct_3.html#D11" onmouseover="rs('rfunct_3/D11')" onmouseout="rh()">RFUNCT_3:def 11</a></span>;<br/>
<a NAME="E33:50"/>
(  <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> is <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp;  <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font></span>)</span> is <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="txt" href="#E13:50"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="mesfun11.html#T10" onmouseover="rs('mesfun11/T10')" onmouseout="rh()">MESFUN11:10</a></span>;<br/>
<a NAME="E34:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A22</font></span>: 
(  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> is <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp;  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> is <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun14.html#T43" target="_self" onmouseover="rs('mesfun14/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E35:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A23</font></span>: 
(  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font> is <font color="Maroon" title="c6">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  &amp;  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font> is <font color="Maroon" title="c6">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a></span>;<br/>
<a NAME="E36:50"/><span class="kw">then </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"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:50"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="ref" href="mesfunc6.html#T61" onmouseover="rs('mesfunc6/T61')" onmouseout="rh()">MESFUNC6:61</a>, <a class="ref" href="mesfunc6.html#T82" onmouseover="rs('mesfunc6/T82')" onmouseout="rh()">MESFUNC6:82</a></span>;<br/>
<a NAME="E37:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A24</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"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="mesfun14.html#T43" target="_self" onmouseover="rs('mesfun14/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">E3</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="E39:50"/><span class="lab"><font color="Green" title="E68">A25</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E3</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E3</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="mesfun14.html#T42" target="_self" onmouseover="rs('mesfun14/T42')" onmouseout="rh()">Th42</a>, <a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</a></span>;<br/>
<a NAME="E40:50"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c10">E3</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E3</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<a NAME="E41:50"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c10">E3</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c10">E3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:50"><span class="lab"><font color="Green" title="E68">A25</font></span></a>, <a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<a NAME="E42:50"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c10">E3</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c10">E3</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c10">E3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E43:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A26</font></span>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c10">E3</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c10">E3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E44:50"/>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</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="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c10">E3</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:50"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="txt" href="#E34:50"><span class="lab"><font color="Green" title="E65">A22</font></span></a>, <a class="txt" href="#E39:50"><span class="lab"><font color="Green" title="E68">A25</font></span></a>, <a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a>, <a class="ref" href="mesfunc6.html#T89" onmouseover="rs('mesfunc6/T89')" onmouseout="rh()">MESFUNC6:89</a></span>;<br/>
<a NAME="E45:50"/><span class="kw">then </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"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</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="c3">M</font>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:50"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E32:50"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="txt" href="#E34:50"><span class="lab"><font color="Green" title="E65">A22</font></span></a>, <a class="txt" href="#E39:50"><span class="lab"><font color="Green" title="E68">A25</font></span></a>, <a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a>, <a class="ref" href="mesfunc6.html#T89" onmouseover="rs('mesfunc6/T89')" onmouseout="rh()">MESFUNC6:89</a></span>;<br/>
<a NAME="E46:50"/><span class="kw">then </span>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:50"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="txt" href="#E35:50"><span class="lab"><font color="Green" title="E66">A23</font></span></a>, <a class="txt" href="#E37:50"><span class="lab"><font color="Green" title="E67">A24</font></span></a>, <a class="ref" href="mesfunc6.html#T61" onmouseover="rs('mesfunc6/T61')" onmouseout="rh()">MESFUNC6:61</a>, <a class="ref" href="mesfunc6.html#T82" onmouseover="rs('mesfunc6/T82')" onmouseout="rh()">MESFUNC6:82</a></span>;<br/>
<a NAME="E47:50"/><span class="kw">then </span>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun14.html#T43" target="_self" onmouseover="rs('mesfun14/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E48:50"/>
<font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R2" title="MESFUNC6: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="#E13:50"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E31:50"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="mesfunc5.html#D17" onmouseover="rs('mesfunc5/D17')" onmouseout="rh()">MESFUNC5:def 17</a>, <a class="ref" href="mesfunc6.html#D4" onmouseover="rs('mesfunc6/D4')" onmouseout="rh()">MESFUNC6:def 4</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">  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <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="c3">M</font>,<font color="Maroon" title="c5">g</font>)</span><br/>

<a NAME="E49:50"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">E0</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c8">E0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E50:50"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c8">E0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<a NAME="E51:50"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">E0</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c8">E0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:50"><span class="lab"><font color="Green" title="E60">A17</font></span></a>, <a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<a NAME="E52:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A27</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c8">E0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E53:50"/>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <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="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c8">E0</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="#E21:50"><span class="lab"><font color="Green" title="E60">A17</font></span></a>, <a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</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="E54:50"/>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <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="c3">M</font>,<font color="Maroon" title="c5">g</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E21:50"><span class="lab"><font color="Green" title="E60">A17</font></span></a>, <a class="txt" href="#E52:50"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="#E12:50"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="mesfun14.html#T26" target="_self" onmouseover="rs('mesfun14/T26')" onmouseout="rh()">Th26</a>, <a class="ref" href="mesfunc6.html#T89" onmouseover="rs('mesfunc6/T89')" onmouseout="rh()">MESFUNC6:89</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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