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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E21">Lm83</font></span>: <a NAME="T15"><span class="comment"><font color="firebrick">:: DUALSP05:15</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">rho</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b3">f</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="dualsp01.html#K17" title="DUALSP01:func.17">DualSp</a> <span class="p2">(<span class="default"><a href="c0sp2.html#K5" title="C0SP2:func.5">R_Normed_Algebra_of_ContinuousFunctions</a> <span class="p3">(<span class="default"><a href="dualsp05.html#K1" title="DUALSP05:func.1">ClstoCmp</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span></span>)</span>  st <font color="Olive" title="b2">rho</font> is  <a href="integr22.html#V1" title="INTEGR22:attr.1">bounded_variation</a>  &amp; (  for <font color="Olive" title="b4">u</font> being   <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr22.html#K5" title="INTEGR22:func.5">integral</a> (<font color="Olive" title="b4">u</font>,<font color="Olive" title="b2">rho</font>) ) holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">f</font></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="integr22.html#K2" title="INTEGR22:func.2">total_vd</a> <font color="Olive" title="b2">rho</font></div></div>
