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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">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><br/> for <font color="Olive" title="b4">Sq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font></span>)</span>  ex <font color="Olive" title="b5">Fsq</font> being  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>for <font color="Olive" title="b6">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b5">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</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="b3">k</font> &amp; <font color="Olive" title="b5">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b6">n</font> &amp; <font color="Olive" title="b4">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font></span>)</span>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font> &amp;  ex <font color="Olive" title="b7">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b7">r</font> &amp; <font color="Olive" title="b7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Olive" title="b2">M</font>,<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="Olive" title="b5">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b3">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b6">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Olive" title="b3">k</font></span>)</span> ) )</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">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><br/> for <font color="Olive" title="b3">Sq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font></span>)</span>  ex <font color="Olive" title="b4">Fsq</font> being  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>for <font color="Olive" title="b5">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b4">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</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="b2">k</font> &amp; <font color="Olive" title="b4">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b5">n</font> &amp; <font color="Olive" title="b3">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font> &amp;  ex <font color="Olive" title="b6">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b6">r</font> &amp; <font color="Olive" title="b6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Olive" title="b1">M</font>,<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="Olive" title="b4">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b2">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Olive" title="b2">k</font></span>)</span> ) )</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">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><br/> for <font color="Olive" title="b2">Sq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font></span>)</span>  ex <font color="Olive" title="b3">Fsq</font> being  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>for <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b3">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</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="Olive" title="b3">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">n</font> &amp; <font color="Olive" title="b2">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font> &amp;  ex <font color="Olive" title="b5">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b5">r</font> &amp; <font color="Olive" title="b5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="Olive" title="b3">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Olive" title="b1">k</font></span>)</span> ) )</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Sq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>  ex <font color="Olive" title="b2">Fsq</font> being  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b2">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</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="c4">k</font> &amp; <font color="Olive" title="b2">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b1">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp;  ex <font color="Olive" title="b4">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b4">r</font> &amp; <font color="Olive" title="b4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="Olive" title="b2">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">Sq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>; <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">Fsq</font> being  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</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="c4">k</font> &amp; <font color="Olive" title="b1">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp;  ex <font color="Olive" title="b3">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b3">r</font> &amp; <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="Olive" title="b1">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> ) )</span><br/>









<span class="kw">consider </span><font color="Maroon" title="c6">Fsq</font> being   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</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="E2:91"/><span class="lab"><font color="Green" title="E70">A1</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</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="c4">k</font> &amp; <font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp;  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T63" target="_self" title="LPSPACE2:th.63">Th004X</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">DMFSQN</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/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">DMFSQN</font> &amp;  ex <font color="Olive" title="b2">FSQN</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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1 <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">FSQN</font> &amp; <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">DMFSQN</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="Olive" title="b2">FSQN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">DMFSQN</font> &amp; <font color="Olive" title="b2">FSQN</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">DMFSQN</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b2">FSQN</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> ) );<br/>
<a NAME="E3:91"/><span class="lab"><font color="Green" title="E71">A2</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E1:91_1"/>
<font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c7">n</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="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E70">A1</font></span></a></span>;<br/>
<a NAME="E2:91_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">FMF</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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c7">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">FMF</font> &amp;  ex <font color="Olive" title="b2">ND</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">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>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">FMF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">ND</font> &amp; <font color="Olive" title="b1">FMF</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b2">ND</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b1">FMF</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">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">hence </span><a NAME="E3:91_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Olive" title="b1">y</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><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E5:91"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c7">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S10" title="FUNCT_2:sch.10">FUNCT_2:sch 10</a>(<a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E71">A2</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c8">E0</font> =  <a href="measure2.html#K1" title="MEASURE2:func.1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span> 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="E7:91"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <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="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span></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="c8">E0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span></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="c8">E0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> )</span><br/>

<a NAME="E1:91_2"/>
 ex <font color="Olive" title="b1">D</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="c7">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> &amp;  ex <font color="Olive" title="b2">F</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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">F</font> &amp; <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">D</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="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</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="b1">D</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">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="#E5:91"><span class="lab"><font color="Green" title="E72">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:91_2"/>
( <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="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span></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="c8">E0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</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>
<a NAME="E8:91"/><span class="lab"><font color="Green" title="E74">B1</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T42" title="MEASURE1:th.42">MEASURE1:42</a></span>;<br/>
<a NAME="E9:91"/>
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span> holds <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">A</font> be    <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"> ( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span> implies ( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_3"/><span class="lab"><font color="Green" title="E75">B11</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</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="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">A0</font> = <font color="Maroon" title="c9">A</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:91_3"/>
<font color="Maroon" title="c10">A0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_3"><span class="lab"><font color="Green" title="E75">B11</font></span></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:91_3"/><span class="lab"><font color="Green" title="E76">B12</font></span>: 
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c10">A0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">n</font> = <font color="Maroon" title="c11">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91_3"><span class="lab"><font color="Green" title="E76">B12</font></span></a></span>;<br/>
<a NAME="E7:91_3"/><span class="lab"><font color="Green" title="E77">B13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">A0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A0</font>
 
;<br/>
<a NAME="E8:91_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">A0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91_3"><span class="lab"><font color="Green" title="E76">B12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:91_3"/>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>; <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="c9">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E10:91_3"/><span class="lab"><font color="Green" title="E78">B14</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">A0</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="txt" href="#E7:91"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E5:91_3"><span class="lab"><font color="Green" title="E76">B12</font></span></a>, <a class="txt" href="#E7:91_3"><span class="lab"><font color="Green" title="E77">B13</font></span></a></span>;<br/>
<a NAME="E11:91_3"/>
<font color="Maroon" title="c10">A0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91_3"><span class="lab"><font color="Green" title="E76">B12</font></span></a>, <a class="txt" href="#E7:91_3"><span class="lab"><font color="Green" title="E77">B13</font></span></a></span>;<br/>
<a NAME="E12:91_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">A</font> 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#T66" title="MEASURE1:th.66">MEASURE1:66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:91_3"/>
<font color="Maroon" title="c9">A</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="#E10:91_3"><span class="lab"><font color="Green" title="E78">B14</font></span></a>, <a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</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>
<a NAME="E10:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">B2</font></span>: 
( ( for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> ) &amp; ( for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span> holds <br/><font color="Olive" title="b1">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> ) )
 
;<br/>
<a NAME="E11:91"/><span class="kw">then </span>
<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E12:91"/><span class="kw">then </span>
<font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span> is    <a href="measure2.html#M1" title="MEASURE2:mode.1">N_Measure_fam</a> of <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:91"><span class="lab"><font color="Green" title="E74">B1</font></span></a>, <a class="ref" href="measure2.html#D1" title="MEASURE2:def.1">MEASURE2:def 1</a></span>;<br/>
<a NAME="E13:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">B3</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span></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="txt" href="#E10:91"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="measure2.html#T16" title="MEASURE2:th.16">MEASURE2:16</a></span>;<br/>
<font color="Maroon" title="c8">E0</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p4">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T15" title="MEASURE1:th.15">MEASURE1:15</a></span>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p3">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T48" title="XBOOLE_1:th.48">XBOOLE_1:48</a></span>
<br/>.= 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="measure1.html#NK1" title="MEASURE1:NK.1">\</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E15:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">B4</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">E0</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="txt" href="#E13:91"><span class="lab"><font color="Green" title="E76">B3</font></span></a>, <a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">Fsq2</font> = <font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font>;<br/>
<a NAME="E16:91"/><span class="lab"><font color="Green" title="E78">A5</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <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"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E0</font></span><br/>

<a NAME="E1:91_5"/>
 <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"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">E0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun9c.html#D1" title="MESFUN9C:def.1">MESFUN9C:def 1</a></span>;<br/>
<a NAME="E2:91_5"/><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"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">E0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:91_5"/>
 <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"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">n</font>, <font color="Maroon" title="c11">m</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span><br/><a NAME="E1:91_6"/>
( <font color="Maroon" title="c10">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c11">m</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E2:91_6"/><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"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E0</font> &amp;  <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"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:91"><span class="lab"><font color="Green" title="E78">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:91_6"/>
 <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"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font></span>)</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Fsq2</font> = <font color="Maroon" title="c6">Fsq</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c8">E0</font> as   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</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="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">Fsq2</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">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</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="c4">k</font> &amp; <font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp;  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</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="c4">k</font> &amp; <font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp;  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> ) )</span><br/><a NAME="E1:91_7"/>
<font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</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="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E70">A1</font></span></a></span>;<br/><a NAME="E2:91_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A7</font></span>: 
 ex <font color="Olive" title="b1">FMF</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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">FMF</font> &amp;  ex <font color="Olive" title="b2">ND</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">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>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">FMF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">ND</font> &amp; <font color="Olive" title="b1">FMF</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b2">ND</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b1">FMF</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">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">reconsider </span><font color="Maroon" title="c12">E2</font> =  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> 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="txt" href="#E2:91_7"><span class="lab"><font color="Green" title="E79">A7</font></span></a></span>;<br/><a NAME="E4:91_7"/><span class="lab"><font color="Green" title="E80">A8</font></span>: 
<font color="Maroon" title="c12">E2</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">E0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E5:91_7"/>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c12">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_7"><span class="lab"><font color="Green" title="E79">A7</font></span></a>, <a class="ref" href="mesfunc6.html#D6" title="MESFUNC6:def.6">MESFUNC6:def 6</a></span>;<br/><a NAME="E6:91_7"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c8">E0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="ref" href="mesfunc1.html#T34" title="MESFUNC1:th.34">MESFUNC1:34</a></span>;<br/><a NAME="E7:91_7"/><span class="kw">then </span>
<font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c8">E0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#D6" title="MESFUNC6:def.6">MESFUNC6:def 6</a></span>;<br/><a NAME="E8:91_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">E0</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c8">E0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_7"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="ref" href="mesfunc6.html#T76" title="MESFUNC6:th.76">MESFUNC6:76</a></span>;<br/><a NAME="E9:91_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A9</font></span>: 
<font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c8">E0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun9c.html#D1" title="MESFUN9C:def.1">MESFUN9C:def 1</a></span>;<br/><a NAME="E10:91_7"/><span class="lab"><font color="Green" title="E82">A10</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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:91"><span class="lab"><font color="Green" title="E78">A5</font></span></a></span>;<br/><a NAME="E11:91_7"/>
(  <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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</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> <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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> &amp;  <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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</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> <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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> )
 <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="E12:91_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A11</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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> &amp;  <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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</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="E13:91_7"/>
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> <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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> holds <br/><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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <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> <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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <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"> ( <font color="Maroon" title="c13">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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> implies <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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_7_1"/><span class="lab"><font color="Green" title="E84">L1</font></span>: 
<font color="Maroon" title="c13">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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">x0</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:91_7_1"/><span class="lab"><font color="Green" title="E85">L2</font></span>: 
<font color="Maroon" title="c13">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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91_7"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="txt" href="#E12:91_7"><span class="lab"><font color="Green" title="E83">A11</font></span></a>, <a class="txt" href="#E4:91_7"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="txt" href="#E1:91_7_1"><span class="lab"><font color="Green" title="E84">L1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">thus </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> <span class="p3">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_7_1"><span class="lab"><font color="Green" title="E84">L1</font></span></a>, <a class="ref" href="mesfun6c.html#D6" title="MESFUN6C:def.6">MESFUN6C:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c4">k</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>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">E0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun9c.html#D1" title="MESFUN9C:def.1">MESFUN9C:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91_7"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="txt" href="#E12:91_7"><span class="lab"><font color="Green" title="E83">A11</font></span></a>, <a class="txt" href="#E1:91_7_1"><span class="lab"><font color="Green" title="E84">L1</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c4">k</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>
<br/>.= 
<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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_7_1"><span class="lab"><font color="Green" title="E85">L2</font></span></a>, <a class="ref" href="mesfun6c.html#D6" title="MESFUN6C:def.6">MESFUN6C:def 6</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><a NAME="E14:91_7"/><span class="kw">then </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> <span class="p3">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">E0</font> <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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:91"><span class="lab"><font color="Green" title="E78">A5</font></span></a>, <a class="txt" href="#E4:91_7"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="txt" href="#E12:91_7"><span class="lab"><font color="Green" title="E83">A11</font></span></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>;<br/><a NAME="E15:91_7"/><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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">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="#E2:91_7"><span class="lab"><font color="Green" title="E79">A7</font></span></a>, <a class="ref" href="mesfunc6.html#T91" title="MESFUNC6:th.91">MESFUNC6:91</a></span>;<br/><span class="kw">hence </span><a NAME="E16:91_7"/><span class="lab"><font color="Green" title="E84">A12</font></span>: 
<font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</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="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:91_7"><span class="lab"><font color="Green" title="E81">A9</font></span></a>, <a class="txt" href="#E10:91_7"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="txt" href="#E15:91"><span class="lab"><font color="Green" title="E77">B4</font></span></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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp;  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> ) )</span><br/><a NAME="E17:91_7"/><span class="lab"><font color="Green" title="E85">A13</font></span>: 
( <font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<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="#E2:91"><span class="lab"><font color="Green" title="E70">A1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">EB</font> = <font color="Maroon" title="c8">E0</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="E19:91_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">EB</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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91_7"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></span>;<br/><a NAME="E20:91_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">EB</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="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">EB</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="mesfun9c.html#D1" title="MESFUN9C:def.1">MESFUN9C:def 1</a></span>;<br/><a NAME="E21:91_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A14</font></span>: 
<font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:91"><span class="lab"><font color="Green" title="E77">B4</font></span></a>, <a class="ref" href="lpspace1.html#D10" title="LPSPACE1:def.10">LPSPACE1:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E22:91_7"/>
<font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:91_7"><span class="lab"><font color="Green" title="E85">A13</font></span></a>, <a class="txt" href="#E16:91_7"><span class="lab"><font color="Green" title="E84">A12</font></span></a>, <a class="ref" href="lpspace2.html#T36" target="_self" title="LPSPACE2:th.36">EQC00b</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="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp;  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> ) )</span><br/><a NAME="E23:91_7"/>
 <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Fsq</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T42" target="_self" title="LPSPACE2:th.42">EQC02bx</a>, <a class="txt" href="#E21:91_7"><span class="lab"><font color="Green" title="E86">A14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E24:91_7"/>
<font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<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="#E2:91"><span class="lab"><font color="Green" title="E70">A1</font></span></a></span>; <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">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> )</span><br/><span class="kw">hence </span><a NAME="E25:91_7"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><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="c10">Fsq2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Sq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T54" target="_self" title="LPSPACE2:th.54">Lm17x</a>, <a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a>, <a class="txt" href="#E16:91_7"><span class="lab"><font color="Green" title="E84">A12</font></span></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>


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