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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K15" title="NEWTON:func.15">|^</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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K15" title="NEWTON:func.15">|^</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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K15" title="NEWTON:func.15">|^</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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
( <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 )
 <span class="kw">and </span><br/><a NAME="E2:70"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<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#K15" title="NEWTON:func.15">|^</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><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#K15" title="NEWTON:func.15">|^</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="E3:70"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<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#K15" title="NEWTON:func.15">|^</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#K15" title="NEWTON:func.15">|^</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="asympt_0.html#T36" target="_self" title="ASYMPT_0:th.36">Th36</a></span>;<br/>
<a NAME="E4:70"/><span class="kw">then </span>
<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#K15" title="NEWTON:func.15">|^</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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="asympt_0.html#T39" target="_self" title="ASYMPT_0:th.39">Th39</a></span>;<br/>
<a NAME="E6:70"/>
<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#K15" title="NEWTON:func.15">|^</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:70"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="asympt_0.html#T38" target="_self" title="ASYMPT_0:th.38">Th38</a></span>;<br/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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