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<b>let </b><font color="Maroon">a</font> be  <a href="asympt_0.html#V1">logbase</a> <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</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#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="power.html#K6">log</a> <font color="Maroon">a</font>,<font color="Olive">n</font> ) )
 ;<br/>

<a NAME="E3:3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="asympt_0.html#D3">ASYMPT_0:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">N</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K2">\]</a></span>;<br/>
<a NAME="E4:3"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<a NAME="E5:3"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">N</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K2">\]</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_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="E1:3_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:3_1"/>
<font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:3_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:3_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="power.html#K6">log</a> <font color="Maroon">a</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> <font color="Maroon">a</font>,<font color="Maroon">N</font>
 <b>by </b><i>, , <a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E5:3_1"/>
 <a href="power.html#K6">log</a> <font color="Maroon">a</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">a</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E6:3_1"/><b>then </b>
 <a href="power.html#K6">log</a> <font color="Maroon">a</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="power.html#T60">POWER:60</a>, </i>;<br/><b>hence </b><a NAME="E7:3_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:3"/>
<font color="Maroon">f</font> is <a href="asympt_0.html#V4">eventually-positive</a>
 <b>by </b><i><a class="ref" href="asympt_0.html#D6">ASYMPT_0:def 6</a></i>;<br/>


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