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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E52">Th42</font></span>: <a NAME="T49"><span class="comment"><font color="firebrick">:: COUSIN2:49</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">I</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">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">jauge</font> being   <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">I</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b4">D</font> being    <a href="cousin.html#M1" title="COUSIN:mode.1">tagged_division</a> of <font color="Olive" title="b1">I</font>  st <font color="Olive" title="b3">jauge</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font>)</span>)</span> &amp; <font color="Olive" title="b4">D</font> is <font color="Olive" title="b3">jauge</font> <a href="cousin.html#V1" title="COUSIN:attr.1">-fine</a>  holds <br/> <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <span class="p1">(<span class="default"><a href="cousin.html#K4" title="COUSIN:func.4">division_of</a> <font color="Olive" title="b4">D</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font></div></div>
