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<div><span class="kw">theorem </span><a NAME="T41"><span class="comment"><font color="firebrick">:: ASYMPT_1:41</font></span><br/></a><div class="add"> ex <font color="Olive" title="b1">F</font> being    <a href="funct_2.html#M1" title="FUNCT_2:mode.1">FUNCTION_DOMAIN</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; (  for <font color="Olive" title="b2">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>  holds  <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> ) &amp;  not  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">to_power</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K6" title="ASYMPT_0:func.6">Big_Oh</a> <span class="p2">(<span class="default"><a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> 1</span>)</span></span>)</span> )</div></div>
