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<span class="kw">take </span>
2
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="asympt_0.html#D4" title="ASYMPT_0:def.4">ASYMPT_0:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></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>   holds <br/>(  not 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c1">g</font> =  <a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">f</font> =  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1;<br/>
<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E1:106"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:106"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E3:106"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A3</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="lab"><font color="Green" title="E66">A4</font></span>: <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> = 
<font color="Maroon" title="c3">n</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c3">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p3">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c3">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:106"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E1:106"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="seq_1.html#T10" title="SEQ_1:th.10">SEQ_1:10</a></span>
<br/>.= 
<font color="Maroon" title="c3">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1

;<br/>
<span class="kw">hence </span><a NAME="E6:106"/>
 not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 1</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" title="ASYMPT_1:func.4">seq_const</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:106"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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