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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:72"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="asympt_0.html#V7" target="_self">smooth</a>
 <b>and </b><br/><a NAME="E2:72"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>and </b><br/><a NAME="E3:72"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K14" target="_self">Big_Theta</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> ;<br/>
<a NAME="E4:72"/>
<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">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">c<sub>1</sub></font>,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> </span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E3:72"><i><font color="Green">E38</font></i></a>, <a class="ref" href="asympt_0.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E5:72"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K12" target="_self">Big_Oh</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K13" target="_self">Big_Omega</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">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">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E1:72"><i><font color="Green">E36</font></i></a>, <a class="txt" href="asympt_0.html#E2:72"><i><font color="Green">E37</font></i></a>, <a class="ref" href="asympt_0.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E7:72"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10" target="_self">Big_Omega</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E1:72"><i><font color="Green">E36</font></i></a>, <a class="txt" href="asympt_0.html#E2:72"><i><font color="Green">E37</font></i></a>, <a class="txt" href="asympt_0.html#E5:72"><i><font color="Green">E39</font></i></a>, <a class="ref" href="asympt_0.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E8:72"/><b>then </b>
<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E6:72"><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">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11" target="_self">Big_Theta</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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