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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E55">Th83</font></span>: <a NAME="T83"><span class="comment"><font color="firebrick">:: DBLSEQ_3:83</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">f</font> being   <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b2">seq</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>  st (  for <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Olive" title="b2">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">m</font>)</span>)</span> ) holds <br/> <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Sum</a> <font color="Olive" title="b2">seq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <span class="p1">(<span class="default"><a href="dblseq_3.html#K8" title="DBLSEQ_3:func.8">lim_in_cod2</a> <span class="p2">(<span class="default"><a href="dblseq_3.html#K9" title="DBLSEQ_3:func.9">Partial_Sums_in_cod2</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span></div></div>
