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<div>:: <span class="kw">deftheorem </span>   defines <a href="integr22.html#V1" title="INTEGR22:attr.1">bounded_variation</a> <a onclick="hs(this)" href="javascript:()">INTEGR22:def 3 : <br/></a><span> 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> holds <br/> ( <font color="Olive" title="b2">rho</font> is  <a href="integr22.html#V1" title="INTEGR22:attr.1">bounded_variation</a>  iff  ex <font color="Olive" title="b3">d</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">d</font> &amp; (  for <font color="Olive" title="b4">t</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">A</font><br/>  for <font color="Olive" title="b5">F</font> being    <a href="integr22.html#M1" title="INTEGR22:mode.1">var_volume</a> of <font color="Olive" title="b2">rho</font>,<font color="Olive" title="b4">t</font> holds   <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b5">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">d</font> ) ) );<br/></span></div>
