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<div><span class="kw">theorem </span><a NAME="T50"><span class="comment"><font color="firebrick">:: SERIES_3:50</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">s1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b2">s1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Olive" title="b1">s</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) ) holds <br/> for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <font color="Olive" title="b2">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font></div></div>
