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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">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> holds <br/> ( <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is   <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> &amp; <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="b3">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="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">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> holds <br/> ( <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is   <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> &amp; <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="b2">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="firebrick">::  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> holds <br/> ( <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is   <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> &amp; <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is   <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> &amp; <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="c4">k</font>) )</span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T46" title="FUNCOP_1:th.46">FUNCOP_1:46</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">ND</font> = <font color="Maroon" title="c1">X</font> 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#T34" title="MEASURE1:th.34">MEASURE1:34</a></span>;<br/>
<a NAME="E3:28"/>
<font color="Maroon" title="c6">ND</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/>
<a NAME="E4:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A1</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="c6">ND</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/>
<a NAME="E5:28"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <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="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:28"/>
 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>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> holds <br/><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</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="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<a NAME="E7:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="lpspace2.html#T15" target="_self" title="LPSPACE2:th.15">Th15</a></span>;<br/>
<a NAME="E8:28"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
;<br/>
<a NAME="E9:28"/><span class="kw">then </span>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T14" target="_self" title="LPSPACE2:th.14">Th14</a>, <a class="ref" href="mesfunc6.html#T52" title="MESFUNC6:th.52">MESFUNC6:52</a></span>;<br/>
<a NAME="E10:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A4</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="c5">g</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T12" target="_self" title="LPSPACE2:th.12">Th12</a></span>;<br/>
<a NAME="E11:28"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> holds <br/><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</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="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<a NAME="E12:28"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">ND</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="lpspace1.html#T52" title="LPSPACE1:th.52">LPSPACE1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:28"/>
( <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is   <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> &amp; <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="c4">k</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E29">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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