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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#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 <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>) &amp; <font color="Olive" title="b4">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="Olive" title="b2">M</font>,<font color="Olive" title="b5">k</font>) holds <br/><font color="Olive" title="b3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b4">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="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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b4">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 <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>) &amp; <font color="Olive" title="b3">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="Olive" title="b1">M</font>,<font color="Olive" title="b4">k</font>) holds <br/><font color="Olive" title="b2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b3">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="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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b3">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 <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>) &amp; <font color="Olive" title="b2">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="Olive" title="b3">k</font>) holds <br/><font color="Olive" title="b1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">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="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="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>  st <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>) &amp; <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="Olive" title="b1">k</font>) holds <br/><font color="Maroon" title="c4">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <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="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="firebrick">::  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>) &amp; <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>) implies <font color="Maroon" title="c4">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <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><br/>











<span class="kw">set </span><font color="Maroon" title="c7">W</font> =  <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>);<br/>
<span class="kw">assume </span><a NAME="E1:30"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
( <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>) &amp; <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>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">f1</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="E3:30"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
( <font color="Maroon" title="c8">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font> &amp;  ex <font color="Olive" title="b1">Ef1</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">Ef1</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="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Ef1</font> &amp; <font color="Maroon" title="c8">f1</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">Ef1</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="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">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="c9">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="E5:30"/><span class="lab"><font color="Green" title="E30">A3</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="c9">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="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Ef</font> &amp; <font color="Maroon" title="c8">f1</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c9">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="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">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="#E3:30"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">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="E7:30"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
( <font color="Maroon" title="c10">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font> &amp;  ex <font color="Olive" title="b1">Eg1</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">Eg1</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="c10">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Eg1</font> &amp; <font color="Maroon" title="c10">g1</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">Eg1</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">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="#E1:30"><span class="lab"><font color="Green" title="E28">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">Eg</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="E9:30"/><span class="lab"><font color="Green" title="E32">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="c11">Eg</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="c10">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Eg</font> &amp; <font color="Maroon" title="c10">g1</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c11">Eg</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">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="#E7:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a></span>;<br/>
<a NAME="E10:30"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c11">Eg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Efg</font> = <font color="Maroon" title="c9">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c11">Eg</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">s</font> = <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">t</font> = <span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><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="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E11:30"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c11">Eg</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">Ef</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c11">Eg</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T54" title="XBOOLE_1:th.54">XBOOLE_1:54</a></span>;<br/>
<a NAME="E12:30"/>
( <font color="Maroon" title="c9">Ef</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c11">Eg</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="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="E13:30"/><span class="kw">then </span>
( <font color="Maroon" title="c9">Ef</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  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="c11">Eg</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  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="#E5:30"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="measure1.html#D7" title="MEASURE1:def.7">MEASURE1:def 7</a></span>;<br/>
<a NAME="E14:30"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">Ef</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Eg</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> 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#T37" title="MEASURE1:th.37">MEASURE1:37</a></span>;<br/>
<a NAME="E15:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A8</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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c11">Eg</font></span>)</span> <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="txt" href="#E11:30"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="measure1.html#D7" title="MEASURE1:def.7">MEASURE1:def 7</a></span>;<br/>
<a NAME="E16:30"/>
( <font color="Maroon" title="c8">f1</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c9">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c11">Eg</font> &amp; <font color="Maroon" title="c10">g1</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c9">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c11">Eg</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="mesfunc6.html#T16" title="MESFUNC6:th.16">MESFUNC6:16</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E17:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c9">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c11">Eg</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T26" title="MESFUNC6:th.26">MESFUNC6:26</a></span>;<br/>
<a NAME="E18:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A10</font></span>: 
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c9">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c11">Eg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="mesfunc6.html#T48" title="MESFUNC6:th.48">MESFUNC6:48</a></span>;<br/>
<a NAME="E19:30"/>
<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="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="lpspace2.html#T22" target="_self" title="LPSPACE2:th.22">Th22</a></span>;<br/>
<a NAME="E20:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><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="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T102" title="MESFUNC6:th.102">MESFUNC6:102</a></span>;<br/>
<a NAME="E21:30"/><span class="lab"><font color="Green" title="E39">A12</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="c8">f1</font></span>)</span> <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="c8">f1</font> &amp;  <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="c10">g1</font></span>)</span> <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="c10">g1</font> &amp;  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/>
<a NAME="E22:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">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="c9">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c11">Eg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E18:30"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="mesfun6c.html#T29" title="MESFUN6C:th.29">MESFUN6C:29</a></span>;<br/>
<a NAME="E23:30"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T14" target="_self" title="LPSPACE2:th.14">Th14</a></span>;<br/>
<a NAME="E24:30"/><span class="lab"><font color="Green" title="E42">A15</font></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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">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="c9">Ef</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c11">Eg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E21:30"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="mesfun6c.html#D4" title="MESFUN6C:def.4">MESFUN6C:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E43">A16</font></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">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><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="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</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"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<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="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c8">f1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#D4" title="MESFUN6C:def.4">MESFUN6C:def 4</a></span>
<br/>.= 
<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="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c8">f1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#D4" title="MESFUN6C:def.4">MESFUN6C:def 4</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:30"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
 <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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:30"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="mesfun6c.html#D4" title="MESFUN6C:def.4">MESFUN6C:def 4</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><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="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:30_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><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="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span><br/><a NAME="E2:30_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</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> <span class="p4">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><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="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="ref" href="lpspace2.html#T24" target="_self" title="LPSPACE2:th.24">Th24</a>, <a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E24:30"><span class="lab"><font color="Green" title="E42">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:30_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><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="c8">f1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c10">g1</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E27:30"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">g1</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">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="#E22:30"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E24:30"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E25:30"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E20:30"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="mesfunc6.html#T96" title="MESFUNC6:th.96">MESFUNC6:96</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:30"/>
<font color="Maroon" title="c4">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <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="#E3:30"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E15:30"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E10:30"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E17:30"><span class="lab"><font color="Green" title="E36">A9</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/>


</div>
