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<b>let </b><font color="Maroon" title="c1">f</font> be  <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/> for <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 <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> &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="c1">f</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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> ex <font color="Olive" title="b3">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b3">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="b4">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="b4">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="Olive" title="b1">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">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="b4">n</font></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c2">t</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="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> &amp; ( for <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 <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">N</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="b2">n</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>  ) holds <br/> ex <font color="Olive" title="b2">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">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="b1">N</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="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">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> ) )</span><br/><b>let </b><font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></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> &amp; ( 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="c3">N</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">n</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>  ) implies  ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="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 <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N</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="b2">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="b2">n</font></span>)</span> ) ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:28"/><i><font color="Green" title="E16">A1</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green" title="E17">A2</font></i>: 
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="c3">N</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">n</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="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 <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N</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="b2">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="b2">n</font></span>)</span> ) )</span><br/>

<b>consider </b><font color="Maroon" title="c4">s</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:28"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">s</font>
 <b>and </b><br/><a NAME="E5:28"/><i><font color="Green" title="E19">A4</font></i>: 
 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>  ) ) )
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:28"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">c2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E7:28"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c5">c2</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> 
 <b>and </b><br/><a NAME="E8:28"/><i><font color="Green" title="E21">A6</font></i>: 
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">M</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">c2</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="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>  )
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:28"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="asympt_0.html#E5:28"><i><font color="Green" title="E19">A4</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1;<br/>
<b>set </b><font color="Maroon" title="c7">tset</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) 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>  : ( <font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">N</font> ) </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1;<br/>
<b>set </b><font color="Maroon" title="c8">fset</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) 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>  : ( <font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">N</font> ) </span>}</span> ;<br/>
<a NAME="E9:28"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E10:28"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E11:28"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) 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>  : ( <font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">N</font> ) </span>}</span>  is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>from </b><i><a class="ref" href="asympt_0.html#S1" target="_self" title="ASYMPT_0:sch.1">ASYMPT_0:sch 1</a>(<a class="txt" href="asympt_0.html#E10:28"><i><font color="Green" title="E23">A8</font></i></a>);<br/></i>
<b>then reconsider </b><font color="Maroon" title="c9">tset</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) 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>  : ( <font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">N</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E13:28"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) 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>  : ( <font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">N</font> ) </span>}</span>  is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>from </b><i><a class="ref" href="asympt_0.html#S1" target="_self" title="ASYMPT_0:sch.1">ASYMPT_0:sch 1</a>(<a class="txt" href="asympt_0.html#E10:28"><i><font color="Green" title="E23">A8</font></i></a>);<br/></i>
<b>then reconsider </b><font color="Maroon" title="c10">fset</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) 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>  : ( <font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">N</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>set </b><font color="Maroon" title="c11">T</font> =  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font>;<br/>
<b>set </b><font color="Maroon" title="c12">F</font> =  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font>;<br/>
<a NAME="E15:28"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">fset</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">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> <b> such that </b><br/><a NAME="E17:28"/><i><font color="Green" title="E24">A9</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font> &amp; <font color="Maroon" title="c13">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c13">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">N</font> )
 ;<br/>
<a NAME="E18:28"/><i><font color="Green" title="E25">A10</font></i>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</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> 
 
<b>by </b><i><a class="txt" href="asympt_0.html#E2:28"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="asympt_0.html#E17:28"><i><font color="Green" title="E24">A9</font></i></a></i>;<br/>
<a NAME="E19:28"/><i><font color="Green" title="E26">A11</font></i>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="asympt_0.html#E2:28"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="asympt_0.html#E17:28"><i><font color="Green" title="E24">A9</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">c1</font> = <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">c</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">c2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">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="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="c3">N</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">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> ) )</span><br/>

<b>thus </b><a NAME="E21:28"/>
<font color="Maroon" title="c15">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> 
 <b>by </b><i><a class="txt" href="asympt_0.html#E7:28"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="c3">N</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">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></span><br/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N</font> 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> <font color="Maroon" title="c15">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="c16">n</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E22:28"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <font color="Maroon" title="c15">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="c16">n</font></span>)</span></span><br/>

<a NAME="E23:28"/><b>then </b><i><font color="Green" title="E28">A13</font></i>: 
<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> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="asympt_0.html#E2:28"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<a NAME="E24:28"/><i><font color="Green" title="E29">A14</font></i>: 
<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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="asympt_0.html#E2:28"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="asympt_0.html#E22:28"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( <font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">M</font> or <font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>suppose </b></a><a NAME="E1:28_1_1"/>
<font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <font color="Maroon" title="c15">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="c16">n</font></span>)</span></span><br/><div class="add"><a NAME="E2:28_1_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">M</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E22:28"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:28_1_1"/><b>then </b><i><font color="Green" title="E30">A15</font></i>: 
<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="c5">c2</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="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:28"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/><a NAME="E4:28_1_1"/>
<font color="Maroon" title="c5">c2</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="c16">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">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="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E23:28"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><b>hence </b><a NAME="E5:28_1_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> <font color="Maroon" title="c15">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="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:28_1_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><a NAME="E1:28_1_2"/><i><font color="Green" title="E30">A16</font></i>: 
<font color="Maroon" title="c3">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <font color="Maroon" title="c15">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="c16">n</font></span>)</span></span><br/><div class="add"><b>thus </b><a NAME="E2:28_1_2"/>
<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="c15">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="c16">n</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_2_1_1"/>
( <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> or <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">M</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_2_1_1_1"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <font color="Maroon" title="c15">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="c16">n</font></span>)</span></span><br/><div class="add"><a NAME="E2:28_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E31">A17</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:28"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:28_1_2_1_1_1"/><b>then </b>
<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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">tset</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E22:28"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/><a NAME="E4:28_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E32">A18</font></i>: 
<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>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E5:28_1_2_1_1_1"/><i><font color="Green" title="E33">A19</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">M</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> 
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:28"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/><a NAME="E6:28_1_2_1_1_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">tset</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E9:28"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="asympt_0.html#E1:28_1_2"><i><font color="Green" title="E30">A16</font></i></a></i>;<br/><a NAME="E7:28_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E34">A20</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/><div><i><font color="Green" title="E35">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:28_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</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 href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E18:28"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:28_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_0.html#E19:28"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="asympt_0.html#E5:28_1_2_1_1_1"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="asympt_0.html#E7:28_1_2_1_1_1"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E9:28_1_2_1_1_1"/>
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">fset</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E22:28"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="asympt_0.html#E2:28_1_2_1_1_1"><i><font color="Green" title="E31">A17</font></i></a></i>;<br/><a NAME="E10:28_1_2_1_1_1"/><b>then </b>
<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> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/><a NAME="E11:28_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E36">A22</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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> <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"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:28_1_2_1_1_1"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><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></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:28_1_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="c16">n</font></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="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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>
 <b>by </b><i><a class="txt" href="asympt_0.html#E23:28"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E3:28_1_2_1_1_1_2"/><b>then </b>
<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#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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>
 <b>by </b><i><a class="txt" href="asympt_0.html#E24:28"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><a NAME="E4:28_1_2_1_1_1_2"/><b>then </b>
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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>
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:28_1_2_1_1_1"><i><font color="Green" title="E32">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:28_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_0.html#E19:28"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="asympt_0.html#E11:28_1_2_1_1_1"><i><font color="Green" title="E36">A22</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E13:28_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="c16">n</font></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="c1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n</font></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"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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>
 <b>by </b><i><a class="txt" href="asympt_0.html#E23:28"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E14:28_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E37">A23</font></i>: 
<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"><span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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>
 <b>by </b><i><a class="txt" href="asympt_0.html#E24:28"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><a NAME="E15:28_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">tset</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c10">fset</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">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="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E23:28"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><b>hence </b><a NAME="E16:28_1_2_1_1_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> <font color="Maroon" title="c15">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="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E14:28_1_2_1_1_1"><i><font color="Green" title="E37">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_2_1_1_2"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <font color="Maroon" title="c15">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="c16">n</font></span>)</span></span><br/><div class="add"><a NAME="E2:28_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E31">A24</font></i>: 
<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="c5">c2</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="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E8:28"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/><a NAME="E3:28_1_2_1_1_2"/>
<font color="Maroon" title="c5">c2</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="c16">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">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="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E23:28"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><b>hence </b><a NAME="E4:28_1_2_1_1_2"/>
<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="c15">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="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:28_1_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
