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<b>let </b><font color="Maroon">c1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<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> ;<br/>
<b>thus </b><a NAME="E3:98_1_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> )
 ;<br/>

<b>assume </b><a NAME="E4:98_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E5:98_1_2"/><i><font color="Green">E59</font></i>: 
(  ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">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>  st <br/>( <font color="Olive">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Olive">l</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <font color="Olive">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="Olive">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="Olive">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="Olive">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> ) ) ) &amp;  ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">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>  st <br/>( <font color="Olive">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">prob28</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Olive">l</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <font color="Olive">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="Olive">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="Olive">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="Olive">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> ) ) ) &amp; not <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">l1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">prob281</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="E7:98_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">prob281</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Maroon">l1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E8:98_1_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">prob281</font> <a href="asympt_1.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">singlea</font> &amp; ( 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">prob281</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Olive">n</font>,<font color="Maroon">prob281</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>] ) )
 ;<br/>
<b>consider </b><font color="Maroon">l2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">prob282</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="E10:98_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">l2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">prob282</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Maroon">l2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E11:98_1_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">prob282</font> <a href="asympt_1.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">singlea</font> &amp; ( 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">prob282</font> <a href="asympt_1.html#K6" target="_self">.</a> <font color="Olive">n</font>,<font color="Maroon">prob282</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><a class="txt" href="asympt_1.html#E8:98_1_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E12:98_1_2"/>
<font color="Maroon">prob281</font> <a href="hidden.html#R1">=</a> <font color="Maroon">prob282</font>
 
<b>from </b><i><a class="ref" href="recdef_1.html#S10">RECDEF_1:sch 10</a>(<a class="txt" href="asympt_1.html#E11:98_1_2"><i><font color="Green">E63</font></i></a>, , <a class="txt" href="asympt_1.html#E7:98_1_2"><i><font color="Green">E60</font></i></a>);<br/></i>
<b>hence </b><a NAME="E13:98_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:98_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="asympt_1.html#E10:98_1_2"><i><font color="Green">E62</font></i></a>, </i>;<br/>


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