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<b>let </b><font color="Maroon">f</font> be  <a href="asympt_0.html#V2" target="_self">eventually-nonnegative</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">t</font> be  <a href="asympt_0.html#V2" target="_self">eventually-nonnegative</a> <a href="asympt_0.html#V6" target="_self">eventually-nondecreasing</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:71"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> is <a href="asympt_0.html#V7" target="_self">smooth</a>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>and </b><br/><a NAME="E3:71"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K13" target="_self">Big_Omega</a> <font color="Maroon">f</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> ;<br/>
<a NAME="E4:71"/>
<font color="Maroon">f</font> <a href="asympt_0.html#R2" target="_self">is_smooth_wrt</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E1:71"><i><font color="Green">E36</font></i></a>, , </i>;<br/>
<a NAME="E5:71"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f</font> is <a href="asympt_0.html#V6" target="_self">eventually-nondecreasing</a> &amp; <font color="Maroon">f</font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">f</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">N1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:71"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">N2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:71"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N2</font> holds <br/><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E11:71"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="asympt_0.html#K15" target="_self">taken_every</a> <font color="Maroon">b</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E12:71"/><i><font color="Green">E69</font></i>: 
 ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">N3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:71"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E15:71"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N3</font> holds <br/>( <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:71"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E17:71"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">r</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E18:71"/><i><font color="Green">E75</font></i>: 
 ex <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  holds <br/>( <font color="Olive">d</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">N4</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E20:71"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E21:71"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N4</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  holds <br/>( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="asympt_0.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">N5</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E23:71"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N5</font> holds <br/><font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">N0</font> =  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span>;<br/>
<a NAME="E24:71"/>
(  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N2</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N3</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N4</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E25:71"/><b>then </b><i><font color="Green">E80</font></i>: 
(  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N2</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N3</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N4</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E26:71"/><i><font color="Green">E81</font></i>: 
(  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N5</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E27:71"/><b>then </b><i><font color="Green">E82</font></i>: 
(  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E28:71"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E29:71"/><b>then </b>
<font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E30:71"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span>
 
<b>by </b><i><a class="ref" href="asympt_0.html#D8" target="_self">Def8</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E31:71"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c</font> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E32:71"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E33:71"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">t</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:71_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:71_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:71"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:71_1"/><b>then </b><i><font color="Green">E89</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N2</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N3</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N4</font> )
 <b>by </b><i><a class="ref" href="asympt_0.html#D6" target="_self">Def6</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">n1</font> = <font color="Maroon">b</font> <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/><a NAME="E4:71_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="asympt_0.html#D8" target="_self">Def8</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:71_1"/><b>then </b><i><font color="Green">E121</font></i>: 
 <a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> <font color="Maroon">b</font>,1
 <b>by </b><i>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E6:71_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:71_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#T21">INT_1:21</a></i>;<br/>
<a NAME="E3:71_1_1"/>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E4:71_1_1"/><b>then </b>
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:71_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1
 
;<br/>
<a NAME="E6:71_1_1"/><b>then </b>
 <a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i><a class="ref" href="asympt_0.html#D11" target="_self">Def11</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E7:71_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="asympt_0.html#D10" target="_self">Def10</a>, <a class="ref" href="power.html#T59">POWER:59</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">i</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E8:71_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">b</font> <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">b</font> <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="newton.html#T99">NEWTON:99</a></i>;<br/><b>set </b><font color="Maroon">n2</font> = <font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font>;<br/><a NAME="E10:71_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="asympt_0.html#D7" target="_self">Def7</a>, </i>;<br/><a NAME="E11:71_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#D3">POWER:def 3</a></i>;<br/><a NAME="E12:71_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E13:71_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">b</font> <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><i><font color="Green">E136</font></i>: <font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/><a NAME="E15:71_1"/>
 <a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T52">INT_1:52</a></i>;<br/><a NAME="E16:71_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="txt" href="asympt_0.html#E12"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="asympt_0.html#T2" target="_self">Th2</a>, <a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><a NAME="E17:71_1"/>
<font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K3">to_power</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><span class="p4">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E18:71_1"/><b>then </b>
<font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 ;<br/><a NAME="E19:71_1"/><b>then </b>
<font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K3">to_power</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">b</font>,<font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E20:71_1"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">n</font> <a href="real_1.html#K6">/</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E21:71_1"/>
<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p5">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p5">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p3">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="asympt_0.html#D8" target="_self">Def8</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E22:71_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p5">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p5">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span>
 ;<br/><a NAME="E23:71_1"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p5">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p5">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E24:71_1"/>
<font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N5</font>,<span class="p2">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> 1,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <span class="p5">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p5">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E25:71_1"/><b>then </b>
<font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E26:71_1"/><b>then </b>
<font color="Maroon">n</font> <a href="real_1.html#K6">/</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E27:71_1"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font></span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N3</font>,<font color="Maroon">N4</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E28:71_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N2</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#D6" target="_self">Def6</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E29:71_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:71"><i><font color="Green">E41</font></i></a>, <a class="txt" href="asympt_0.html#E12"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="asympt_0.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E30:71_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="asympt_0.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E31:71_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N4</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#D6" target="_self">Def6</a>, <a class="txt" href="asympt_0.html#E12:71"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E32:71_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="asympt_0.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E33:71_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N3</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#D6" target="_self">Def6</a>, <a class="txt" href="asympt_0.html#E12:71"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E34:71_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E35:71_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E11:71"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E36:71_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>
 ;<br/><a NAME="E37:71_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E38:71_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E5:71"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E39:71_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E40:71_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:71"><i><font color="Green">E40</font></i></a>, , , </i>;<br/><a NAME="E41:71_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E42:71_1"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E14:71"><i><font color="Green">E70</font></i></a>, <a class="txt" href="asympt_0.html#E15:71"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E43:71_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E44:71_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E17:71"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E45:71_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N5</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#D7" target="_self">Def7</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E46:71_1"/>
<font color="Maroon">t</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="asympt_0.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E35:71"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10" target="_self">Big_Omega</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="asympt_0.html#D9" target="_self">Def9</a></i>;<br/>


</div>
