<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E21">Th1920</font></span>: <a NAME="T20"><span class="comment"><font color="firebrick">:: INTEGR21:20</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">Y</font> being   <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</a><br/>  for <font color="Olive" title="b3">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">Y</font>  st <font color="Olive" title="b1">A</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> <font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">A</font> is  <a href="integr19.html#V1" title="INTEGR19:attr.1">bounded</a>  &amp; <font color="Olive" title="b3">f</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Olive" title="b1">A</font> &amp; <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Olive" title="b3">f</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">A</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">A</font>)</span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Olive" title="b3">f</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span>,<font color="Olive" title="b1">A</font>)</div></div>
