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<b>let </b><font color="Maroon">f</font> be  <a href="asympt_0.html#V2" target="_self">eventually-nonnegative</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">t</font> be  <a href="asympt_0.html#V2" target="_self">eventually-nonnegative</a> <a href="asympt_0.html#V6" target="_self">eventually-nondecreasing</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:72"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> is <a href="asympt_0.html#V7" target="_self">smooth</a>
 <b>and </b><br/><a NAME="E2:72"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>and </b><br/><a NAME="E3:72"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K14" target="_self">Big_Theta</a> <font color="Maroon">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> ;<br/>
<a NAME="E4:72"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K12" target="_self">Big_Oh</a> <font color="Maroon">f</font>,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K13" target="_self">Big_Omega</a> <font color="Maroon">f</font>,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:72"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K12" target="_self">Big_Oh</a> <font color="Maroon">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K13" target="_self">Big_Omega</a> <font color="Maroon">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:72"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E1:72"><i><font color="Green">E36</font></i></a>, , </i>;<br/>
<a NAME="E7:72"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10" target="_self">Big_Omega</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E1:72"><i><font color="Green">E36</font></i></a>, , , </i>;<br/>
<a NAME="E8:72"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K10" target="_self">Big_Omega</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E2:72"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:72"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11" target="_self">Big_Theta</a> <font color="Maroon">f</font>
 ;<br/>


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