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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E17">Th9</font></span>: <a NAME="T9"><span class="comment"><font color="firebrick">:: LEIBNIZ1:9</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/>(  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><a href="leibniz1.html#K2" title="LEIBNIZ1:func.2">Leibniz_Series_of</a> <font color="Olive" title="b1">r</font></span>)</span> is  <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative-yielding</a>  &amp;  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><a href="leibniz1.html#K2" title="LEIBNIZ1:func.2">Leibniz_Series_of</a> <font color="Olive" title="b1">r</font></span>)</span> is  <a href="valued_0.html#V8" title="VALUED_0:attr.8">non-increasing</a>  &amp;  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><a href="leibniz1.html#K2" title="LEIBNIZ1:func.2">Leibniz_Series_of</a> <font color="Olive" title="b1">r</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><a href="leibniz1.html#K2" title="LEIBNIZ1:func.2">Leibniz_Series_of</a> <font color="Olive" title="b1">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</div></div>
