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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being   <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a>   <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">f</font> is  <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a>  &amp; <font color="Olive" title="b2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#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="Olive" title="b2">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b3">n</font></span>)</span> where <font color="Olive" title="b3">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span> } </span> ) holds <br/><font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">t</font> be   <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a>   <a href="asympt_0.html#V6" title="ASYMPT_0:attr.6">eventually-nondecreasing</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">f</font> is  <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a>  &amp; <font color="Olive" title="b1">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#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="Olive" title="b1">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b2">n</font></span>)</span> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span> } </span> ) holds <br/><font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K13" title="ASYMPT_0:func.13">Big_Oh</a> (<font color="Maroon" title="c1">f</font>,<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span> } </span> ) implies <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c1">f</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:68"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="asympt_0.html#V7" title="ASYMPT_0:attr.7">smooth</a> 
 <span class="kw">and </span><br/><a NAME="E2:68"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">and </span><br/><a NAME="E3:68"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K13" title="ASYMPT_0:func.13">Big_Oh</a> (<font color="Maroon" title="c1">f</font>,<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span> } </span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c1">f</font></span><br/>

<a NAME="E4:68"/><span class="lab"><font color="Green" title="E37">A4</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="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="asympt_0.html#D17" target="_self" title="ASYMPT_0:def.17">Def20</a></span>;<br/>
<a NAME="E5:68"/><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="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>
 
<span class="kw">by</span> <span class="lab"><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">s</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="E7:68"/><span class="lab"><font color="Green" title="E38">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="c3">b</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">s</font>
 <span class="kw">and </span><br/><a NAME="E8:68"/><span class="lab"><font color="Green" title="E39">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">s</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">s</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">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="E10:68"/><span class="lab"><font color="Green" title="E40">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="E11:68"/><span class="lab"><font color="Green" title="E41">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">N3</font> holds <br/>( <font color="Maroon" title="c4">s</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">s</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="#E8:68"><span class="lab"><font color="Green" title="E39">A6</font></span></a></span>;<br/>
<a NAME="E12:68"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon" title="c3">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="#E2:68"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:68"/>
<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="#E4:68"><span class="lab"><font color="Green" title="E37">A4</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="c7">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> <span class="kw"> such that </span><br/><a NAME="E15:68"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 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="c7">N1</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">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#K1" title="SEQ_1:func.1">.</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>
 <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/>
<span class="kw">consider </span><font color="Maroon" title="c8">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="E17:68"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
 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="c8">N2</font> holds <br/><font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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>
 <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/>
<span class="kw">set </span><font color="Maroon" title="c9">X</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span> } </span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">r</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="E19:68"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E20:68"/><span class="lab"><font color="Green" title="E46">A13</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> &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#K15" title="NEWTON:func.15">|^</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#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="c10">r</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>  ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E22:68"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E23:68"/><span class="lab"><font color="Green" title="E48">A15</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="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#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b2">n</font></span>)</span> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span> } </span>  holds <br/>( <font color="Maroon" title="c10">r</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="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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c10">r</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="#E20:68"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>);<br/>
<a NAME="E24:68"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
 <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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c7">N1</font>,<font color="Maroon" title="c8">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="E25:68"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c7">N1</font>,<font color="Maroon" title="c8">N2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">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="E26:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A17</font></span>: 
 <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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c8">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:68"><span class="lab"><font color="Green" title="E49">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="E27:68"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c7">N1</font>,<font color="Maroon" title="c8">N2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">N1</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="E28:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A18</font></span>: 
 <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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c7">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:68"><span class="lab"><font color="Green" title="E49">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="E29:68"/><span class="lab"><font color="Green" title="E52">A19</font></span>: 
 <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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c6">N3</font>,<font color="Maroon" title="c12">N4</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="E30:68"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">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>
 
<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="E31:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A20</font></span>: 
 <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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:68"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E32:68"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c6">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="c6">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/>
<a NAME="E33:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A21</font></span>: 
 <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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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">N3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:68"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">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> <span class="kw"> such that </span><br/><a NAME="E35:68"/><span class="lab"><font color="Green" title="E55">A22</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="c14">N5</font> holds <br/><font color="Maroon" title="c2">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="ref" href="asympt_0.html#D2" target="_self" title="ASYMPT_0:def.2">Def4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">N</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>);<br/>
<a NAME="E36:68"/><span class="lab"><font color="Green" title="E56">A23</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)</span>)</span></span>)</span>)
 
<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="E37:68"/>
 <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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)</span>)</span>
 
<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="E38:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A24</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:68"><span class="lab"><font color="Green" title="E56">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E39:68"/>
<font color="Maroon" title="c3">b</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="#E2:68"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E40:68"/><span class="kw">then </span>
<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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)</span>)</span>
 
<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="E41:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A25</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:68"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E42:68"/><span class="lab"><font color="Green" title="E59">A26</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c14">N5</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="E43:68"/><span class="lab"><font color="Green" title="E60">A27</font></span>: 
 <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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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
 
<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="E44:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A28</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:68"><span class="lab"><font color="Green" title="E56">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E62">A29</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:68_1"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:68"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E2:68_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)</span>)</span>
 ;<br/><a NAME="E3:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A30</font></span>: 
<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> (<font color="Maroon" title="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>;<br/><span class="kw">let </span><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> ; <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="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="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>) implies ( <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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="c5">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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> &amp; <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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>  ) )</span><br/><span class="kw">set </span><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/><span class="kw">assume </span><a NAME="E4:68_1"/><span class="lab"><font color="Green" title="E64">A31</font></span>: 
<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="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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="c5">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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> &amp; <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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>  )</span><br/><a NAME="E5:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A32</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:68"><span class="lab"><font color="Green" title="E56">A23</font></span></a>, <a class="txt" href="#E43:68"><span class="lab"><font color="Green" title="E60">A27</font></span></a>, <a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="power.html#D3" title="POWER:def.3">POWER:def 3</a></span>;<br/><a NAME="E6:68_1"/>
<font color="Maroon" title="c16">n</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="#E44:68"><span class="lab"><font color="Green" title="E61">A28</font></span></a>, <a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:68_1"/><span class="kw">then </span>
 <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)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></span>;<br/><a NAME="E8:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A33</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="power.html#T51" title="POWER:th.51">POWER:51</a></span>;<br/><div><span class="lab"><font color="Green" title="E67">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:68_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D6" title="INT_1:def.6">INT_1:def 6</a></span>;<br/><a NAME="E2:68_1_1"/><span class="kw">then </span>
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:68_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A35</font></span>: 
<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/><span class="kw">assume </span><a NAME="E4:68_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#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="E5:68_1_1"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T8" title="INT_1:th.8">INT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E6:68_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68_1"><span class="lab"><font color="Green" title="E66">A33</font></span></a>, <a class="txt" href="#E3:68_1_1"><span class="lab"><font color="Green" title="E68">A35</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</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">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><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="c19">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68_1"><span class="lab"><font color="Green" title="E67">A34</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><a NAME="E12:68_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#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T41" title="POWER:th.41">POWER:41</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">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/><span class="kw">set </span><font color="Maroon" title="c21">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="c20">n1</font>;<br/><span class="lab"><font color="Green" title="E68">A36</font></span>: <font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">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>
<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="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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></span>
;<br/><a NAME="E15:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c19">n3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T41" title="POWER:th.41">POWER:41</a></span>;<br/><a NAME="E16:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A37</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">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#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span> } </span> 
 ;<br/><a NAME="E17:68_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="c14">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E18:68_1"/><span class="kw">then </span>
<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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E19:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A38</font></span>: 
<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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>;<br/><a NAME="E20:68_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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T29" title="INT_1:th.29">INT_1:29</a></span>;<br/><a NAME="E21:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A39</font></span>: 
<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="c20">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E5:68_1"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="txt" href="#E14:68_1"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="ref" href="pre_ff.html#T8" title="PRE_FF:th.8">PRE_FF:8</a></span>;<br/><a NAME="E22:68_1"/><span class="kw">then </span>
<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>
 <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="E23:68_1"/><span class="kw">then </span>
<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="E24:68_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>;<br/><a NAME="E25:68_1"/><span class="kw">then </span>
<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="c20">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>;<br/><a NAME="E26:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A40</font></span>: 
<font color="Maroon" title="c20">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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:68_1"><span class="lab"><font color="Green" title="E70">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E27:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A41</font></span>: 
<font color="Maroon" title="c20">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">N3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:68"><span class="lab"><font color="Green" title="E54">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E28:68_1"/><span class="lab"><font color="Green" title="E74">A42</font></span>: 
<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="c7">N1</font>,<font color="Maroon" title="c8">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="c6">N3</font>,<font color="Maroon" title="c12">N4</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:68"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E29:68_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:68"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E30:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">N4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:68_1"><span class="lab"><font color="Green" title="E71">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E31:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A43</font></span>: 
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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="c20">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#K1" title="SEQ_1:func.1">.</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="c20">n1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:68"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="txt" href="#E23:68"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E16:68_1"><span class="lab"><font color="Green" title="E69">A37</font></span></a></span>;<br/><a NAME="E32:68_1"/>
<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="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="asympt_0.html#D15" target="_self" title="ASYMPT_0:def.15">Def18</a></span>;<br/><a NAME="E33:68_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="c3">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">n1</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="c20">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E27:68_1"><span class="lab"><font color="Green" title="E73">A41</font></span></a></span>;<br/><a NAME="E34:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A44</font></span>: 
<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#K1" title="SEQ_1:func.1">.</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="c20">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="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="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">n1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:68"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E35:68_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="c8">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:68"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="txt" href="#E28:68_1"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E36:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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="c20">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:68"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E21:68_1"><span class="lab"><font color="Green" title="E71">A39</font></span></a>, <a class="ref" href="asympt_0.html#T1" target="_self" title="ASYMPT_0:th.1">Th1</a></span>;<br/><a NAME="E37:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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="c20">n1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:68_1"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E38:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A45</font></span>: 
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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="c5">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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:68_1"><span class="lab"><font color="Green" title="E76">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E39:68_1"/><span class="lab"><font color="Green" title="E78">A46</font></span>: 
<font color="Maroon" title="c20">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:68"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="txt" href="#E26:68_1"><span class="lab"><font color="Green" title="E72">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E40:68_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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D6" title="INT_1:def.6">INT_1:def 6</a></span>;<br/><a NAME="E41:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E5:68_1"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="ref" href="pre_ff.html#T8" title="PRE_FF:th.8">PRE_FF:8</a></span>;<br/><a NAME="E42:68_1"/><span class="kw">then </span>
<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="c5">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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">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="c5">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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:68"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E22:68"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E39:68_1"><span class="lab"><font color="Green" title="E78">A46</font></span></a>, <a class="ref" href="asympt_0.html#T1" target="_self" title="ASYMPT_0:th.1">Th1</a>, <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="E43:68_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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="c5">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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:68_1"><span class="lab"><font color="Green" title="E77">A45</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"> <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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> </span><br/><a NAME="E44:68_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="c14">N5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:68"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E45:68_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:68"><span class="lab"><font color="Green" title="E55">A22</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="E46:68"/><span class="lab"><font color="Green" title="E63">A47</font></span>: 
<font color="Maroon" title="c2">t</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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<a NAME="E47:68"/>
<font color="Maroon" title="c11">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="#E10:68"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E22:68"><span class="lab"><font color="Green" title="E47">A14</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="E48:68"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:68"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="txt" href="#E45:68"><span class="lab"><font color="Green" title="E62">A29</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>
