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<b>reconsider </b><font color="Maroon">singlea</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">prob28</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:98_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">singlea</font>
 <b>and </b><br/><a NAME="E4:98_1_1"/><i><font color="Green">E51</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Olive">n</font>,<font color="Maroon">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:98_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 implies  ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">c</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>assume </b><a NAME="E6:98_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:98_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> 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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Maroon">l</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
;<br/>

<b>take </b>
<font color="Maroon">l</font>
;<br/>

<b>take </b>
<font color="Maroon">prob28</font>
;<br/>

<b>thus </b><a NAME="E10:98_1_1"/>
( <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Maroon">l</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Maroon">l</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> &amp; <font color="Maroon">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) )
 <b>by </b><i>, , </i>;<br/>


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