<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E57">Th55</font></span>: <a NAME="T55"><span class="comment"><font color="firebrick">:: LPSPACE2:55</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b3">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b5">k</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b6">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b3">M</font>,<font color="Olive" title="b5">k</font>)</span>)</span>  st <font color="Olive" title="b4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">x</font> holds <br/>( <font color="Olive" title="b6">x</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> (<font color="Olive" title="b4">f</font>,<font color="Olive" title="b3">M</font>,<font color="Olive" title="b5">k</font>) &amp;  ex <font color="Olive" title="b7">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">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="b3">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> <font color="Olive" title="b4">f</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Olive" title="b5">k</font></span>)</span>) &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">r</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">1 <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <font color="Olive" title="b5">k</font></span>)</span> ) )</div></div>
