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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E42">Th40</font></span>: <a NAME="T40"><span class="comment"><font color="firebrick">:: MESFUNC5:40</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>, <font color="Olive" title="b5">g</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#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Olive" title="b4">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b5">g</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Olive" title="b2">S</font> &amp; (  for <font color="Olive" title="b6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b6">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"><font color="Olive" title="b4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Olive" title="b5">g</font></span>)</span> holds <br/><font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">x</font> ) holds <br/><font color="Olive" title="b4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Olive" title="b5">g</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> </div></div>
