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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>;<br/><b>let </b><font color="Maroon" title="c2">t</font> be  <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a> <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a>
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>and </b><br/><a NAME="E3:68"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K13" title="ASYMPT_0:func.13">Big_Oh</a> <font color="Maroon" title="c1">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon" title="c4">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> ;<br/>
<a NAME="E4:68"/>
<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="c3">b</font>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E1:68"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="asympt_0.html#E2:68"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="asympt_0.html#D20" target="_self" title="ASYMPT_0:def.20">Def20</a></i>;<br/>
<a NAME="E5:68"/><b>then </b><i><font color="Green" title="E37">A4</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="c3">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="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="c5">N1</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="E7:68"/><i><font color="Green" title="E38">A5</font></i>: 
for <font color="Olive" title="b1">m</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">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N1</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</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#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D8" target="_self" title="ASYMPT_0:def.8">Def8</a></i>;<br/>
<b>consider </b><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> <b> such that </b><br/><a NAME="E9:68"/><i><font color="Green" title="E39">A6</font></i>: 
for <font color="Olive" title="b1">m</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">m</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="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D8" target="_self" title="ASYMPT_0:def.8">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">s</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="E11:68"/><i><font color="Green" title="E40">A7</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="c3">b</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">s</font>
 <b>and </b><br/><a NAME="E12:68"/><i><font color="Green" title="E41">A8</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="c7">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Maroon" title="c7">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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>  ) ) )
 <b>by </b><i><a class="txt" href="asympt_0.html#E5:68"><i><font color="Green" title="E37">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c9">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="E14:68"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c8">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="E15:68"/><i><font color="Green" title="E43">A10</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="c9">N3</font> holds <br/>( <font color="Maroon" title="c7">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="c8">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#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c7">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#E12:68"><i><font color="Green" title="E41">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">r</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="E17:68"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c10">r</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">t</font>
 <b>and </b><br/><a NAME="E18:68"/><i><font color="Green" title="E45">A12</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> &amp; <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b4">n</font></span>)</span> where <font color="Olive" title="b4">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  holds <br/>( <font color="Maroon" title="c10">r</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Maroon" title="c10">r</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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>  ) ) )
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:68"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c12">N4</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="E20:68"/><i><font color="Green" title="E46">A13</font></i>: 
<font color="Maroon" title="c11">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>and </b><br/><a NAME="E21:68"/><i><font color="Green" title="E47">A14</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="c12">N4</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b2">n</font></span>)</span> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  holds <br/>( <font color="Maroon" title="c10">r</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="c11">a</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#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c10">r</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#E18:68"><i><font color="Green" title="E45">A12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">N5</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="E23:68"/><i><font color="Green" title="E48">A15</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="c13">N5</font> holds <br/><font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="ref" href="asympt_0.html#D4" target="_self" title="ASYMPT_0:def.4">Def4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">N0</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span>;<br/>
<a NAME="E24:68"/>
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N1</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N2</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">N3</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">N4</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span> <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">N1</font>,<font color="Maroon" title="c6">N2</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span> <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="c9">N3</font>,<font color="Maroon" title="c12">N4</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E25:68"/><b>then </b><i><font color="Green" title="E49">A16</font></i>: 
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N1</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N2</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">N3</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">N4</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">N</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c13">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E26:68"/><i><font color="Green" title="E50">A17</font></i>: 
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c13">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">N5</font> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c13">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></span>)</span> <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> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E27:68"/><b>then </b><i><font color="Green" title="E51">A18</font></i>: 
(  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c13">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c13">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E28:68"/><i><font color="Green" title="E52">A19</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c3">b</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; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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#E2:68"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E29:68"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span>
 
<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="E30:68"/><b>then </b><i><font color="Green" title="E53">A20</font></i>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c13">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E27:68"><i><font color="Green" title="E51">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="E31:68"/><i><font color="Green" title="E54">A21</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">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#E14:68"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="asympt_0.html#E20:68"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E32:68"/><i><font color="Green" title="E55">A22</font></i>: 
<font color="Maroon" title="c2">t</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">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> ;<br/><b>assume </b><a NAME="E1:68_1"/><i><font color="Green" title="E56">A23</font></i>: 
<font color="Maroon" title="c16">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="c13">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:68_1"/><b>then </b>
<font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E30:68"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:68_1"/><b>then </b><i><font color="Green" title="E57">A24</font></i>: 
( <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N1</font> &amp; <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N2</font> &amp; <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">N3</font> &amp; <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">N4</font> )
 <b>by </b><i><a class="txt" href="asympt_0.html#E25:68"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c17">n1</font> = <font color="Maroon" title="c3">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/><a NAME="E4:68_1"/>
<font color="Maroon" title="c16">n</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#E27:68"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="asympt_0.html#E1:68_1"><i><font color="Green" title="E56">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:68_1"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font> <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="c3">b</font>,1
 <b>by </b><i><a class="txt" href="asympt_0.html#E28:68"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></i>;<br/><a NAME="E6:68_1"/><b>then </b><i><font color="Green" title="E58">A25</font></i>: 
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">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#E28:68"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></i>;<br/><a NAME="E7:68_1"/><i><font color="Green" title="E59">A26</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_1_1_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:68_1_1_1"/><b>then </b><i><font color="Green" title="E60">A27</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T21" title="INT_1:th.21">INT_1:21</a></i>;<br/><a NAME="E3:68_1_1_1"/>
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E4:68_1_1_1"/><b>then </b>
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:68_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ;<br/><b>hence </b><a NAME="E6:68_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_0.html#E6:68_1"><i><font color="Green" title="E58">A25</font></i></a>, <a class="txt" href="asympt_0.html#E2:68_1_1_1"><i><font color="Green" title="E60">A27</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:68_1_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <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/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c18">i</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</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="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E9:68_1"/>
<font color="Maroon" title="c3">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:68"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="power.html#T48" title="POWER:th.48">POWER:48</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">n1</font> = <font color="Maroon" title="c3">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</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>  ;<br/><b>set </b><font color="Maroon" title="c20">n2</font> = <font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font>;<br/><a NAME="E11:68_1"/><i><font color="Green" title="E60">A28</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E26:68"><i><font color="Green" title="E50">A17</font></i></a>, <a class="txt" href="asympt_0.html#E28:68"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="asympt_0.html#E1:68_1"><i><font color="Green" title="E56">A23</font></i></a>, <a class="ref" href="power.html#D3" title="POWER:def.3">POWER:def 3</a></i>;<br/><a NAME="E12:68_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E13:68_1"/><b>then </b><i><font color="Green" title="E61">A29</font></i>: 
<font color="Maroon" title="c19">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E28:68"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="asympt_0.html#E11:68_1"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></i>;<br/><i><font color="Green" title="E62">A30</font></i>: <font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></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="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="asympt_0.html#E2:68"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
;<br/><a NAME="E15:68_1"/>
 <a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T52" title="INT_1:th.52">INT_1:52</a></i>;<br/><a NAME="E16:68_1"/><b>then </b><i><font color="Green" title="E63">A31</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E28:68"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="asympt_0.html#E11:68_1"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="asympt_0.html#E14:68_1"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></i>;<br/><a NAME="E17:68_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </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="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</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="c3">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span>
 <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="E18:68_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </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="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">b</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="c3">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 ;<br/><a NAME="E19:68_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:68"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E20:68_1"/><b>then </b><i><font color="Green" title="E64">A32</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">n1</font>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E21:68_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:68_1"><i><font color="Green" title="E59">A26</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c21">n3</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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="c3">b</font>,<font color="Maroon" title="c16">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 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="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E23:68_1"/>
<font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c21">n3</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:68"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="asympt_0.html#E14:68_1"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="power.html#T48" title="POWER:th.48">POWER:48</a></i>;<br/><a NAME="E24:68_1"/><b>then </b><i><font color="Green" title="E65">A33</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 ;<br/><a NAME="E25:68_1"/><i><font color="Green" title="E66">A34</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">N4</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:68_1"><i><font color="Green" title="E57">A24</font></i></a>, <a class="txt" href="asympt_0.html#E16:68_1"><i><font color="Green" title="E63">A31</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E26:68_1"/><i><font color="Green" title="E67">A35</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:68"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="asympt_0.html#E3:68_1"><i><font color="Green" title="E57">A24</font></i></a>, <a class="txt" href="asympt_0.html#E16:68_1"><i><font color="Green" title="E63">A31</font></i></a>, <a class="ref" href="asympt_0.html#T1" target="_self" title="ASYMPT_0:th.1">Th1</a></i>;<br/><a NAME="E27:68_1"/><i><font color="Green" title="E68">A36</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">a</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#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E17:68"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="asympt_0.html#E21:68"><i><font color="Green" title="E47">A14</font></i></a>, <a class="txt" href="asympt_0.html#E24:68_1"><i><font color="Green" title="E65">A33</font></i></a>, <a class="txt" href="asympt_0.html#E25:68_1"><i><font color="Green" title="E66">A34</font></i></a></i>;<br/><a NAME="E28:68_1"/><i><font color="Green" title="E69">A37</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n1</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E11:68"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="asympt_0.html#D18" target="_self" title="ASYMPT_0:def.18">Def18</a></i>;<br/><a NAME="E29:68_1"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c13">N5</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p5">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p5">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></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="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</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="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p3">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E27:68"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E30:68_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c13">N5</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p5">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p5">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></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="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span>
 ;<br/><a NAME="E31:68_1"/><b>then </b><i><font color="Green" title="E70">A38</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c13">N5</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p5">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p5">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></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="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <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> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:68"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E32:68_1"/>
<font color="Maroon" title="c16">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c13">N5</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> 1,<span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p5">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p5">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span></span>)</span></span>)</span></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="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:68_1"><i><font color="Green" title="E56">A23</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E33:68_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <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> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E31:68_1"><i><font color="Green" title="E70">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E34:68_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">b</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> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E35:68_1"/><b>then </b><i><font color="Green" title="E71">A39</font></i>: 
<font color="Maroon" title="c19">n1</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> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c5">N1</font>,<font color="Maroon" title="c6">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c9">N3</font>,<font color="Maroon" title="c12">N4</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E20:68_1"><i><font color="Green" title="E64">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E36:68_1"/><b>then </b>
<font color="Maroon" title="c19">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">N3</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E25:68"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E37:68_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E15:68"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="asympt_0.html#E28:68_1"><i><font color="Green" title="E69">A37</font></i></a></i>;<br/><a NAME="E38:68_1"/><b>then </b><i><font color="Green" title="E72">A40</font></i>: 
<font color="Maroon" title="c11">a</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#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E20:68"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E39:68_1"/>
<font color="Maroon" title="c19">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N1</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E25:68"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="asympt_0.html#E35:68_1"><i><font color="Green" title="E71">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E40:68_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n1</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:68"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="asympt_0.html#E13:68_1"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="asympt_0.html#T1" target="_self" title="ASYMPT_0:th.1">Th1</a></i>;<br/><a NAME="E41:68_1"/><b>then </b><i><font color="Green" title="E73">A41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n1</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="c11">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E31:68"><i><font color="Green" title="E54">A21</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E42:68_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">a</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#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">n1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E26:68_1"><i><font color="Green" title="E67">A35</font></i></a>, <a class="txt" href="asympt_0.html#E27:68_1"><i><font color="Green" title="E68">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E43:68_1"/><b>then </b>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">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="c11">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E38:68_1"><i><font color="Green" title="E72">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E44:68_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">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="c11">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c</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#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E41:68_1"><i><font color="Green" title="E73">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E45:68_1"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">N5</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E26:68"><i><font color="Green" title="E50">A17</font></i></a>, <a class="txt" href="asympt_0.html#E1:68_1"><i><font color="Green" title="E56">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E46:68_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">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#E23:68"><i><font color="Green" title="E48">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E34:68"/>
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E31:68"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="asympt_0.html#E32:68"><i><font color="Green" title="E55">A22</font></i></a></i>;<br/>


</div>
