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<div><span class="kw">theorem </span><a NAME="T13"><span class="comment"><font color="firebrick">:: INTEGR25:13</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">d</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">left_closed_halfline</a> <font color="Olive" title="b2">d</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="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Olive" title="b2">d</font> holds <br/> for <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b4">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">d</font> holds <br/>( <font color="Olive" title="b1">f</font> <a href="integr10.html#R1" title="INTEGR10:pred.1">is_right_ext_Riemann_integrable_on</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b1">f</font> <a href="integr10.html#R2" title="INTEGR10:pred.2">is_left_ext_Riemann_integrable_on</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> )</div></div>
