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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"> (  ex <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 <br/>( <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="c1">f</font> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> <font color="Olive" title="b1">b</font> ) implies <font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a> )</span><br/>

<b>given </b><font color="Maroon" title="c2">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> <b> such that </b><a NAME="E2:67"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>and </b><br/><a NAME="E3:67"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> <font color="Maroon" title="c2">b</font>
 ; <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></span><br/>

<a NAME="E4:67"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a> &amp; <font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c2">b</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#E3:67"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="asympt_0.html#D19" target="_self" title="ASYMPT_0:def.19">Def19</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">t</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E6:67"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c2">b</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">t</font>
 <b>and </b><br/><a NAME="E7:67"/><i><font color="Green" title="E37">A5</font></i>: 
 ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b2">N</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 <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b3">n</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="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">N</font> holds <br/>( <font color="Maroon" title="c3">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Maroon" title="c3">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c4">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c5">N2</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> <b> such that </b><br/><a NAME="E9:67"/><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E10:67"/><i><font color="Green" title="E39">A7</font></i>: 
for <font color="Olive" title="b1">n</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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N2</font> holds <br/>( <font color="Maroon" title="c3">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c3">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:67"><i><font color="Green" title="E37">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">N3</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> <b> such that </b><br/><a NAME="E12:67"/><i><font color="Green" title="E40">A8</font></i>: 
for <font color="Olive" title="b1">n</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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N3</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:67"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="asympt_0.html#D8" target="_self" title="ASYMPT_0:def.8">Def8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">N</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N2</font>,<font color="Maroon" title="c6">N3</font>;<br/>
<a NAME="E13:67"/><i><font color="Green" title="E41">A9</font></i>: 
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N2</font>,<font color="Maroon" title="c6">N3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N2</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N2</font>,<font color="Maroon" title="c6">N3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N3</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</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="c8">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 implies <font color="Maroon" title="c1">f</font> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> <font color="Maroon" title="c8">a</font> )</span><br/><b>assume </b><a NAME="E1:67_1"/><i><font color="Green" title="E42">A10</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> <font color="Maroon" title="c8">a</font></span><br/><a NAME="E2:67_1"/><i><font color="Green" title="E43">A11</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><b>set </b><font color="Maroon" title="c9">i</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c8">a</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>;<br/><a NAME="E3:67_1"/><i><font color="Green" title="E44">A12</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:67"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:67_1"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:67"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/><a NAME="E5:67_1"/>
<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:67_1"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:67_1"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,1
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:67_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></i>;<br/><a NAME="E7:67_1"/><b>then </b><i><font color="Green" title="E46">A14</font></i>: 
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:67"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="asympt_0.html#E4:67_1"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></i>;<br/><a NAME="E8:67_1"/><i><font color="Green" title="E47">A15</font></i>: 
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c8">a</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="ref" href="int_1.html#D5" title="INT_1:def.5">INT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">i</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c8">a</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> as    <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>  <b>by </b><i><a class="txt" href="asympt_0.html#E7:67_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E10:67_1"/>
<font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c10">i</font>
 ;<br/><a NAME="E11:67_1"/><b>then </b><i><font color="Green" title="E48">A16</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:67"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">i1</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">i</font> as    <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>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">n</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="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N2</font>,<font color="Maroon" title="c6">N3</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><b>assume </b><a NAME="E1:67_1_1"/><i><font color="Green" title="E49">A17</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N2</font>,<font color="Maroon" title="c6">N3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><a NAME="E2:67_1_1"/><b>then </b><i><font color="Green" title="E50">A18</font></i>: 
( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N2</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N3</font> )
 <b>by </b><i><a class="txt" href="asympt_0.html#E13:67"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:67_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:67_1_1_1"/>
<font color="Maroon" title="c2">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c8">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c10">i</font>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E3:67_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="asympt_0.html#E8:67_1"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></i>;<br/>
<b>hence </b><a NAME="E2:67_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:67"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="asympt_0.html#E4:67_1"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="power.html#D3" title="POWER:def.3">POWER: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/>


</div><b>end;</b></div><a NAME="E4:67_1_1"/><b>then </b><i><font color="Green" title="E51">A20</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E5:67_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:67_1"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:67_1_1"/><b>then </b><i><font color="Green" title="E52">A21</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E7:67_1_1"/><b>then </b>
<font color="Maroon" title="c8">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N3</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:67_1_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:67_1_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E12:67"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="asympt_0.html#E4:67_1_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="asympt_0.html#T1" target="_self" title="ASYMPT_0:th.1">Th1</a></i>;<br/><a NAME="E9:67_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D18" target="_self" title="ASYMPT_0:def.18">Def18</a></i>;<br/><a NAME="E10:67_1_1"/><b>then </b><i><font color="Green" title="E53">A22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#D18" target="_self" title="ASYMPT_0:def.18">Def18</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> $1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> $1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>;<br/><a NAME="E11:67_1_1"/><i><font color="Green" title="E54">A23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_2"><b>proof </b></a><div class="add">

<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span> = 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">n</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>
<br/>.= 
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>
;<br/>
<b>hence </b><a NAME="E2:67_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E12:67_1_1"/><i><font color="Green" title="E55">A24</font></i>: 
for <font color="Olive" title="b1">k</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">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c13">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>set </b><font color="Maroon" title="c14">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font>;<br/>
<b>assume </b><a NAME="E1:67_1_1_3"/><i><font color="Green" title="E56">A25</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<i><font color="Green" title="E57">A26</font></i>: <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span> = 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">n</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c13">k</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">n</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_0.html#E2:67"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span>

;<br/>
<b>reconsider </b><font color="Maroon" title="c15">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font> as    <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>  ;<br/>
<a NAME="E4:67_1_1_3"/>
<font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_3_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_1_3_2_1"/>
( <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:67_1_1_3_2_1_1"/>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N2</font></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">n</font>

<br/>.= 
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font>
<b>by </b><i><a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>
<br/>.= 
<font color="Maroon" title="c12">n</font>

;<br/><b>hence </b><a NAME="E3:67_1_1_3_2_1_1"/>
<font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N2</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E13:67"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="asympt_0.html#E1:67_1_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:67_1_1_3_2_1_2"/>
<font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N2</font></span><br/><div class="add"><a NAME="E2:67_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:67_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="power.html#T40" title="POWER:th.40">POWER:40</a></i>;<br/><a NAME="E3:67_1_1_3_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E4:67_1_1_3_2_1_2"/>
<font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N2</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:67_1_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:67_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E6:67"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="asympt_0.html#E10:67"><i><font color="Green" title="E39">A7</font></i></a></i>;<br/>
<a NAME="E6:67_1_1_3"/><b>then </b><i><font color="Green" title="E58">A27</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="asympt_0.html#D18" target="_self" title="ASYMPT_0:def.18">Def18</a></i>;<br/>
<a NAME="E7:67_1_1_3"/>
<font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E9:67"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="asympt_0.html#E1:67_1_1_3"><i><font color="Green" title="E56">A25</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E8:67_1_1_3"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c13">k</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
;<br/>
<a NAME="E9:67_1_1_3"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c13">k</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/>
<a NAME="E10:67_1_1_3"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E9:67"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>;<br/>
<b>hence </b><a NAME="E11:67_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:67_1_1_3"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="asympt_0.html#E6:67_1_1_3"><i><font color="Green" title="E58">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E13:67_1_1"/>
for <font color="Olive" title="b1">k</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="asympt_0.html#E11:67_1_1"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="asympt_0.html#E12:67_1_1"><i><font color="Green" title="E55">A24</font></i></a>);<br/></i><a NAME="E14:67_1_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 ;<br/><a NAME="E15:67_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D18" target="_self" title="ASYMPT_0:def.18">Def18</a></i>;<br/><b>hence </b><a NAME="E16:67_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E10:67_1_1"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E17:67_1_1"/><i><font color="Green" title="E56">A28</font></i>: 
( <font color="Maroon" title="c3">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> &amp; <font color="Maroon" title="c3">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="asympt_0.html#E10:67"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="asympt_0.html#E2:67_1_1"><i><font color="Green" title="E50">A18</font></i></a></i>;<br/><div><i><font color="Green" title="E57">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_1_1_4"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:67_1_1_4"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:67"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:67_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_0.html#E17:67_1_1"><i><font color="Green" title="E56">A28</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E19:67_1_1"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E12:67"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="asympt_0.html#E2:67_1_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="txt" href="asympt_0.html#E6:67_1_1"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="asympt_0.html#T1" target="_self" title="ASYMPT_0:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E20:67_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="asympt_0.html#E18:67_1_1"><i><font color="Green" title="E57">A29</font></i></a>, <a class="ref" href="asympt_0.html#D18" target="_self" title="ASYMPT_0:def.18">Def18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E14:67_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="asympt_0.html#K16" title="ASYMPT_0:func.16">taken_every</a> <font color="Maroon" title="c8">a</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#E2:67_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="txt" href="asympt_0.html#E11:67_1"><i><font color="Green" title="E48">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E15:67_1"/>
<font color="Maroon" title="c1">f</font> <a href="asympt_0.html#R2" title="ASYMPT_0:pred.2">is_smooth_wrt</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:67"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="asympt_0.html#D19" target="_self" title="ASYMPT_0:def.19">Def19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:67"/>
<font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a>
 <b>by </b><i><a class="ref" href="asympt_0.html#D20" target="_self" title="ASYMPT_0:def.20">Def20</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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