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<b>let </b><font color="Maroon" title="c1">f</font> be  <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a> <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a> &amp; <font color="Olive" title="b2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">Big_Theta</a> <font color="Maroon" title="c1">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b3">n</font></span>)</span> where <font color="Olive" title="b3">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  holds <br/><font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c1">f</font></span><br/><b>let </b><font color="Maroon" title="c2">t</font> be  <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a> <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a> &amp; <font color="Olive" title="b1">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">Big_Theta</a> <font color="Maroon" title="c1">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b2">n</font></span>)</span> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  holds <br/><font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c1">f</font></span><br/><b>let </b><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a> &amp; <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">Big_Theta</a> <font color="Maroon" title="c1">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  implies <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c1">f</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:70"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a>
 <b>and </b><br/><a NAME="E2:70"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>and </b><br/><a NAME="E3:70"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K15" title="ASYMPT_0:func.15">Big_Theta</a> <font color="Maroon" title="c1">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c1">f</font></span><br/>

<b>set </b><font color="Maroon" title="c4">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> ;<br/>
<a NAME="E4:70"/>
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K13" title="ASYMPT_0:func.13">Big_Oh</a> <font color="Maroon" title="c1">f</font>,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K14" title="ASYMPT_0:func.14">Big_Omega</a> <font color="Maroon" title="c1">f</font>,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> </span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E3:70"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="asympt_0.html#T36" target="_self" title="ASYMPT_0:th.36">Th36</a></i>;<br/>
<a NAME="E5:70"/><b>then </b><i><font color="Green" title="E38">A4</font></i>: 
( <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K13" title="ASYMPT_0:func.13">Big_Oh</a> <font color="Maroon" title="c1">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K14" title="ASYMPT_0:func.14">Big_Omega</a> <font color="Maroon" title="c1">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:70"/><b>then </b><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E1:70"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="asympt_0.html#E2:70"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="asympt_0.html#T38" target="_self" title="ASYMPT_0:th.38">Th38</a></i>;<br/>
<a NAME="E7:70"/>
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E1:70"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="asympt_0.html#E2:70"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="asympt_0.html#E5:70"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="asympt_0.html#T39" target="_self" title="ASYMPT_0:th.39">Th39</a></i>;<br/>
<b>hence </b><a NAME="E8:70"/>
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E6:70"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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