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<div><span class="kw">theorem </span><a NAME="T12"><span class="comment"><font color="firebrick">:: HOLDER_1:12</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">ap</font>, <font color="Olive" title="b6">bq</font>, <font color="Olive" title="b7">ab</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st (  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b5">ap</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b8">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b6">bq</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">b</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b8">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b7">ab</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">a</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b8">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">b</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b8">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> ) ) &amp; <font color="Olive" title="b5">ap</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp; <font color="Olive" title="b6">bq</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  holds <br/>( <font color="Olive" title="b7">ab</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp;  <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Olive" title="b7">ab</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Olive" title="b5">ap</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Olive" title="b6">bq</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b2">q</font></span>)</span></span>)</span> )</div></div>
