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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E54">Th40</font></span>: <a NAME="T40"><span class="comment"><font color="firebrick">:: INTEGR25:40</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">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">f</font> <a href="integr25.html#R2" title="INTEGR25:pred.2">is_+infty_improper_integrable_on</a> <font color="Olive" title="b2">a</font> &amp;  <a href="integr25.html#K2" title="INTEGR25:func.2">improper_integral_+infty</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  holds <br/> for <font color="Olive" title="b3">Intf</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>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">Intf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">right_closed_halfline</a> <font color="Olive" title="b2">a</font> &amp; (  for <font color="Olive" title="b4">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b4">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> <font color="Olive" title="b3">Intf</font> holds <br/><font color="Olive" title="b3">Intf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">x</font>) ) holds <br/><font color="Olive" title="b3">Intf</font> is  <a href="limfunc1.html#V5" title="LIMFUNC1:attr.5">divergent_in+infty_to-infty</a> </div></div>
