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<div><span class="kw">theorem </span><a NAME="T1"><span class="comment"><font color="firebrick">:: ASYMPT_1:1</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">t1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b3">n</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>   st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Olive" title="b1">t</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">12 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p5">(<span class="default"><font color="Olive" title="b3">n</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Olive" title="b3">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">5 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Olive" title="b3">n</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 36 ) &amp; (  for <font color="Olive" title="b3">n</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>   st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Olive" title="b2">t1</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> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Olive" title="b3">n</font>)</span>)</span> ) holds <br/> ex <font color="Olive" title="b3">s</font>, <font color="Olive" title="b4">s1</font> being   <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <font color="Olive" title="b3">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b4">s1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">t1</font> &amp; <font color="Olive" title="b3">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="asympt_0.html#K6" title="ASYMPT_0:func.6">Big_Oh</a> <font color="Olive" title="b4">s1</font> )</div></div>
