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<div class="add">

<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"> (  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/>

<span class="kw">given </span><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> <span class="kw"> such that </span><a NAME="E2:67"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">and </span><br/><a NAME="E3:67"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<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:()"><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> </span><br/>

<a NAME="E4:67"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="asympt_0.html#D16" target="_self" title="ASYMPT_0:def.16">Def19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">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> <span class="kw"> such that </span><br/><a NAME="E6:67"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 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="c3">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D6" target="_self" title="ASYMPT_0:def.6">Def8</a></span>;<br/>
<a NAME="E7:67"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="asympt_0.html#D16" target="_self" title="ASYMPT_0:def.16">Def19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">t</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)<span class="kw"> such that </span><br/><a NAME="E9:67"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<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="c4">t</font>
 <span class="kw">and </span><br/><a NAME="E10:67"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 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="c4">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="c4">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/>
<span class="kw">consider </span><font color="Maroon" title="c5">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c6">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> <span class="kw"> such that </span><br/><a NAME="E12:67"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<font color="Maroon" title="c5">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> 
 <span class="kw">and </span><br/><a NAME="E13:67"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
 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">N2</font> holds <br/>( <font color="Maroon" title="c4">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="c5">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="c4">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E38">A6</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">N</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c3">N3</font>);<br/>
<a NAME="E14:67"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c3">N3</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E15:67"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c3">N3</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">set </span><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="E1:67_1"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></span>;<br/><a NAME="E2:67_1"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:67_1"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E33">A1</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E4:67_1"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E5:67_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:67_1"/><span class="kw">then </span>
 <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)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="power.html#T57" title="POWER:th.57">POWER:57</a></span>;<br/><a NAME="E7:67_1"/><span class="kw">then </span>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="power.html#T51" title="POWER:th.51">POWER:51</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">i1</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</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><span class="lab"><font color="Green" title="E47">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c6">N2</font>,<font color="Maroon" title="c3">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="c5">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/><span class="kw">defpred </span><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> ] <span class="kw">means </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="newton.html#K15" title="NEWTON:func.15">|^</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="c5">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="E1:67_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E2:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A16</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E3:67_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="ref" href="pre_ff.html#T8" title="PRE_FF:th.8">PRE_FF:8</a></span>;<br/><a NAME="E4:67_1_1"/><span class="kw">then </span>
<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#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="power.html#D3" title="POWER:def.3">POWER:def 3</a></span>;<br/><a NAME="E5:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A17</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><span class="kw">assume </span><a NAME="E6:67_1_1"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
<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="c6">N2</font>,<font color="Maroon" title="c3">N3</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c5">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="E7:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A19</font></span>: 
<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">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:67"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A20</font></span>: 
<font color="Maroon" title="c4">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="c5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:67"><span class="lab"><font color="Green" title="E40">A8</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E53">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:67_1_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#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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:67_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">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="c5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:67_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:67"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E7:67_1_1"><span class="lab"><font color="Green" title="E51">A19</font></span></a>, <a class="txt" href="#E8:67_1_1"><span class="lab"><font color="Green" title="E52">A20</font></span></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/></div><span class="kw">end;</span></div><a NAME="E10:67_1_1"/><span class="lab"><font color="Green" title="E54">A22</font></span>: 
 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_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><font color="Maroon" title="c14">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</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/>
<span class="kw">assume </span><a NAME="E1:67_1_1_2"/><span class="lab"><font color="Green" title="E55">A23</font></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="newton.html#K15" title="NEWTON:func.15">|^</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="c5">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</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="E3:67_1_1_2"/>
<font color="Maroon" title="c5">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="c5">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="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E1:67_1_1_2"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E4:67_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">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="c5">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="E5:67_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">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="c5">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="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>;<br/>
<a NAME="E6:67_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A24</font></span>: 
<font color="Maroon" title="c5">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="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></span>;<br/>
<a NAME="E7:67_1_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="c6">N2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_1_1_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>  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_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_2_1_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c6">N2</font></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>
<br/>.= 
<font color="Maroon" title="c12">n</font>

;<br/><span class="kw">hence </span><a NAME="E3:67_1_1_2_1_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="c6">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:67"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E6:67_1_1"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_2_1_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c6">N2</font></span><br/><div class="add"><a NAME="E2:67_1_1_2_1_1_2"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="power.html#T35" title="POWER:th.35">POWER:35</a></span>;<br/><a NAME="E3:67_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E4:67_1_1_2_1_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="c6">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67_1_1"><span class="lab"><font color="Green" title="E51">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E8:67_1_1_2"/><span class="kw">then </span>
<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="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E13:67"><span class="lab"><font color="Green" title="E40">A8</font></span></a></span>;<br/>
<a NAME="E9:67_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A25</font></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"><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="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D15" target="_self" title="ASYMPT_0:def.15">Def18</a></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"><font color="Maroon" title="c2">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></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"><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#K15" title="NEWTON:func.15">|^</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></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"><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#K15" title="NEWTON:func.15">|^</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/>
<span class="kw">hence </span><a NAME="E11:67_1_1_2"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67_1_1_2"><span class="lab"><font color="Green" title="E57">A25</font></span></a>, <a class="txt" href="#E6:67_1_1_2"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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/>


</div><span class="kw">end;</span></div><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#K15" title="NEWTON:func.15">|^</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>
<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="c5">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>
;<br/><a NAME="E12:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A26</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E12:67_1_1"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="txt" href="#E10:67_1_1"><span class="lab"><font color="Green" title="E54">A22</font></span></a>);<br/></span><a NAME="E14:67_1_1"/><span class="kw">then </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="newton.html#K15" title="NEWTON:func.15">|^</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="c5">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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A27</font></span>: 
<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="c5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D15" target="_self" title="ASYMPT_0:def.15">Def18</a></span>;<br/><a NAME="E16:67_1_1"/><span class="lab"><font color="Green" title="E57">A28</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:67"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="txt" href="#E6:67_1_1"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:67_1_1"/><span class="kw">then </span>
<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="c3">N3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E18:67_1_1"/><span class="kw">then </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"><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E5:67_1_1"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="asympt_0.html#T1" target="_self" title="ASYMPT_0:th.1">Th1</a></span>;<br/><a NAME="E19:67_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D15" target="_self" title="ASYMPT_0:def.15">Def18</a></span>;<br/><a NAME="E20:67_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D15" target="_self" title="ASYMPT_0:def.15">Def18</a></span>;<br/><span class="kw">hence </span><a NAME="E21:67_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="c5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:67_1_1"><span class="lab"><font color="Green" title="E56">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E22:67_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E16:67_1_1"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="txt" href="#E2:67_1_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="asympt_0.html#T1" target="_self" title="ASYMPT_0:th.1">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E23:67_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67_1_1"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="asympt_0.html#D15" target="_self" title="ASYMPT_0:def.15">Def18</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/></div><span class="kw">end;</span></div><a NAME="E11:67_1"/>
<font color="Maroon" title="c5">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="c5">c</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c10">i</font>
 ;<br/><a NAME="E12:67_1"/><span class="kw">then </span>
( <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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) &amp; <font color="Maroon" title="c5">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E13:67_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_1"><span class="lab"><font color="Green" title="E47">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:67_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="asympt_0.html#D16" target="_self" title="ASYMPT_0:def.16">Def19</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:67"/>
<font color="Maroon" title="c1">f</font> is  <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_0.html#D17" target="_self" title="ASYMPT_0:def.17">Def20</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/>


</div>
