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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  ex <font color="Olive" title="b6">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font> st <br/>( <font color="Olive" title="b2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b6">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="b3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b6">E</font> ) &amp; <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b5">k</font> holds <br/>( <font color="Olive" title="b4">g</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b5">k</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  ex <font color="Olive" title="b5">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b5">E</font> ) &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">M</font>,<font color="Olive" title="b4">k</font> holds <br/>( <font color="Olive" title="b3">g</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b4">k</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  ex <font color="Olive" title="b4">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font> ) &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b3">k</font> holds <br/>( <font color="Olive" title="b2">g</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b3">k</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  ex <font color="Olive" title="b2">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">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="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font> ) &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font> holds <br/>( <font color="Maroon" title="c5">g</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> ) &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font> implies ( <font color="Maroon" title="c5">g</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font> ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> )
 <span class="kw">and </span><br/><a NAME="E2:53"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">g</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font> )</span><br/>

<a NAME="E3:53"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 ex <font color="Olive" title="b1">r</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>  st <br/>( <font color="Maroon" title="c5">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c3">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:53"/>
<font color="Maroon" title="c5">g</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T29" title="LPSPACE1:th.29">LPSPACE1:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font></span><br/>

<a NAME="E5:53"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">g1</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> <span class="kw"> such that </span><br/><a NAME="E7:53"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
( <font color="Maroon" title="c5">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">g1</font> &amp;  ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp; <font color="Maroon" title="c7">g1</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> ) )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">Eh</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="E9:53"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Eh</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Eh</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c8">Eh</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53"><span class="lab"><font color="Green" title="E44">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">ND</font> = <font color="Maroon" title="c8">Eh</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T66" title="MEASURE1:th.66">MEASURE1:66</a></span>;<br/>
<a NAME="E11:53"/>
 ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c10">AFK</font> = <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">AGK</font> = <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">Ef</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="E2:53_1"/><span class="lab"><font color="Green" title="E46">B1</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Ef</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c12">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> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">Ef</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">Ef</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Ef</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ef</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">Ef</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">EE</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="E4:53_1"/><span class="lab"><font color="Green" title="E47">B2</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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="c13">EE</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="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">EE</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="#E3:53"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="lpspace1.html#D10" title="LPSPACE1:def.10">LPSPACE1:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">E1</font> = <font color="Maroon" title="c9">ND</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c13">EE</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E6:53_1"/>
<font color="Maroon" title="c13">EE</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">E1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E7:53_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">EE</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></span>;<br/>
<a NAME="E8:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">B3</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="c14">E1</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"><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="c13">EE</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="c14">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <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="c14">E1</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"><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="c13">EE</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="c14">E1</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="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></span>;<br/>
<a NAME="E9:53_1"/><span class="lab"><font color="Green" title="E49">B4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ef</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E46">B1</font></span></a>, <a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/>
<a NAME="E10:53_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ef</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#D6" title="MESFUN6C:def.6">MESFUN6C:def 6</a></span>;<br/>
<a NAME="E11:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">B5</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ef</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ef</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#D2" title="MESFUNC2:def.2">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3" title="MESFUNC2:def.3">MESFUNC2:def 3</a></span>;<br/>
<a NAME="E12:53_1"/>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">Ef</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E46">B1</font></span></a>, <a class="ref" href="mesfunc6.html#T48" title="MESFUNC6:th.48">MESFUNC6:48</a></span>;<br/>
<a NAME="E13:53_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">Ef</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:53_1"><span class="lab"><font color="Green" title="E49">B4</font></span></a>, <a class="ref" href="mesfun6c.html#T29" title="MESFUN6C:th.29">MESFUN6C:29</a></span>;<br/>
<a NAME="E14:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">B6</font></span>: 
( <font color="Maroon" title="c12">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">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">Ef</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:53_1"><span class="lab"><font color="Green" title="E49">B4</font></span></a>, <a class="ref" href="mesfun6c.html#D6" title="MESFUN6C:def.6">MESFUN6C:def 6</a>, <a class="ref" href="mesfunc6.html#D6" title="MESFUNC6:def.6">MESFUNC6:def 6</a></span>;<br/>
<a NAME="E15:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">B7</font></span>: 
(  <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">Ef</font> &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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">Ef</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T27" title="MESFUNC2:th.27">MESFUNC2:27</a>, <a class="ref" href="mesfunc2.html#T28" title="MESFUNC2:th.28">MESFUNC2:28</a></span>;<br/>
<a NAME="E16:53_1"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T14" title="MESFUNC2:th.14">MESFUNC2:14</a>, <a class="ref" href="mesfunc2.html#T15" title="MESFUNC2:th.15">MESFUNC2:15</a></span>;<br/>
<a NAME="E17:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">B8</font></span>: 
(  <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></span>;<br/>
<a NAME="E18:53_1"/><span class="lab"><font color="Green" title="E54">B9</font></span>: 
<font color="Maroon" title="c12">Ef</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c14">E1</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Ef</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c14">E1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">E0</font> = <font color="Maroon" title="c12">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c14">E1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">E2</font> = <font color="Maroon" title="c12">Ef</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c14">E1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E21:53_1"/>
(  <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span>)</span> &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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></span>;<br/>
<a NAME="E22:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">B10</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p4">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c15">E0</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p4">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span></span>)</span> &amp;  <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p4">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c15">E0</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p4">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53_1"><span class="lab"><font color="Green" title="E50">B5</font></span></a>, <a class="txt" href="#E15:53_1"><span class="lab"><font color="Green" title="E52">B7</font></span></a>, <a class="txt" href="#E17:53_1"><span class="lab"><font color="Green" title="E53">B8</font></span></a>, <a class="txt" href="#E18:53_1"><span class="lab"><font color="Green" title="E54">B9</font></span></a>, <a class="ref" href="xboole_1.html#T89" title="XBOOLE_1:th.89">XBOOLE_1:89</a>, <a class="ref" href="mesfunc5.html#T87" title="MESFUNC5:th.87">MESFUNC5:87</a></span>;<br/>
<a NAME="E23:53_1"/><span class="lab"><font color="Green" title="E56">B11</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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c15">E0</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c15">E0</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:53_1"><span class="lab"><font color="Green" title="E52">B7</font></span></a>, <a class="txt" href="#E17:53_1"><span class="lab"><font color="Green" title="E53">B8</font></span></a>, <a class="txt" href="#E11:53_1"><span class="lab"><font color="Green" title="E50">B5</font></span></a>, <a class="ref" href="mesfunc5.html#T86" title="MESFUNC5:th.86">MESFUNC5:86</a></span>;<br/>
<a NAME="E24:53_1"/>
( <font color="Maroon" title="c9">ND</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c13">EE</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:53"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E4:53_1"><span class="lab"><font color="Green" title="E47">B2</font></span></a>, <a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</a></span>;<br/>
<a NAME="E25:53_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">E1</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T70" title="MEASURE1:th.70">MEASURE1:70</a></span>;<br/>
<a NAME="E26:53_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</a></span>;<br/>
<a NAME="E27:53_1"/><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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53_1"><span class="lab"><font color="Green" title="E50">B5</font></span></a>, <a class="txt" href="#E15:53_1"><span class="lab"><font color="Green" title="E52">B7</font></span></a>, <a class="txt" href="#E17:53_1"><span class="lab"><font color="Green" title="E53">B8</font></span></a>, <a class="ref" href="mesfunc5.html#T88" title="MESFUNC5:th.88">MESFUNC5:88</a></span>;<br/>
<a NAME="E28:53_1"/><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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c15">E0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c15">E0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53_1"><span class="lab"><font color="Green" title="E50">B5</font></span></a>, <a class="txt" href="#E15:53_1"><span class="lab"><font color="Green" title="E52">B7</font></span></a>, <a class="txt" href="#E17:53_1"><span class="lab"><font color="Green" title="E53">B8</font></span></a>, <a class="txt" href="#E23:53_1"><span class="lab"><font color="Green" title="E56">B11</font></span></a>, <a class="ref" href="mesfunc5.html#T89" title="MESFUNC5:th.89">MESFUNC5:89</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E29:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">B12</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span> &amp;  <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:53_1"><span class="lab"><font color="Green" title="E55">B10</font></span></a>, <a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>;<br/>
<a NAME="E30:53_1"/>
<font color="Maroon" title="c12">Ef</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c14">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ef</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">E1</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#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></span>;<br/>
<a NAME="E31:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">B13</font></span>: 
<font color="Maroon" title="c16">E2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E32:53_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font> <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="c14">E1</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> <font color="Maroon" title="c16">E2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_1"><span class="lab"><font color="Green" title="E47">B2</font></span></a>, <a class="txt" href="#E8:53_1"><span class="lab"><font color="Green" title="E48">B3</font></span></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></span>;<br/>
<a NAME="E33:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">B14</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font> <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> <font color="Maroon" title="c16">E2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:53_1"><span class="lab"><font color="Green" title="E58">B13</font></span></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></span>;<br/>
<a NAME="E34:53_1"/><span class="lab"><font color="Green" title="E60">B15</font></span>: 
( <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T62" title="RFUNCT_1:th.62">RFUNCT_1:62</a></span>;<br/>
<a NAME="E35:53_1"/><span class="lab"><font color="Green" title="E61">B16</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <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="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <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="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T20" target="_self" title="LPSPACE2:th.20">Lm001g</a></span>;<br/>
<a NAME="E36:53_1"/><span class="lab"><font color="Green" title="E62">B17</font></span>: 
( <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span> &amp; <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span> &amp; <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span> &amp; <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#T34" title="MESFUNC5:th.34">MESFUNC5:34</a></span>;<br/>
<a NAME="E37:53_1"/><span class="lab"><font color="Green" title="E63">B18</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:53"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="mesfunc6.html#D9" title="MESFUNC6:def.9">MESFUNC6:def 9</a></span>;<br/>
<a NAME="E38:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">B19</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp;  <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></span>;<br/>
<a NAME="E39:53_1"/>
(  <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"><span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> &amp;  <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"><span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">E2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:53_1"><span class="lab"><font color="Green" title="E63">B18</font></span></a>, <a class="ref" href="mesfunc5.html#T103" title="MESFUNC5:th.103">MESFUNC5:103</a></span>;<br/>
<a NAME="E40:53_1"/><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#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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp;  <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:53_1"><span class="lab"><font color="Green" title="E57">B12</font></span></a>, <a class="txt" href="#E33:53_1"><span class="lab"><font color="Green" title="E59">B14</font></span></a>, <a class="txt" href="#E34:53_1"><span class="lab"><font color="Green" title="E60">B15</font></span></a>, <a class="txt" href="#E35:53_1"><span class="lab"><font color="Green" title="E61">B16</font></span></a>, <a class="txt" href="#E36:53_1"><span class="lab"><font color="Green" title="E62">B17</font></span></a>, <a class="txt" href="#E38:53_1"><span class="lab"><font color="Green" title="E64">B19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E41:53_1"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:53_1"><span class="lab"><font color="Green" title="E51">B6</font></span></a>, <a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E42:53_1"/>
( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Ef</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ef</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">Ef</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E46">B1</font></span></a>, <a class="ref" href="mesfunc6.html#D9" title="MESFUNC6:def.9">MESFUNC6:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:53"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
