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<span class="kw">set </span><font color="Maroon" title="c1">g</font> =  <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 3;<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> 2;<br/>
<div><span class="lab"><font color="Green" title="E26">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">now </span></a><div class="add"><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"> ( <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 implies ( <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 2</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> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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> 3</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 2</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><span class="kw">assume </span><a NAME="E1:38_1"/><span class="lab"><font color="Green" title="E27">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"> ( <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 2</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> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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> 3</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 2</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><a NAME="E2:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <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/><a NAME="E3:38_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 2</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> <font color="Maroon" title="c3">n</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></span>;<br/><a NAME="E4:38_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 3</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> <font color="Maroon" title="c3">n</font> <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="asympt_1.html#D3" target="_self" title="ASYMPT_1:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:38_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 2</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> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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> 3</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="txt" href="#E2:38_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 2</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">thus </span><a NAME="E6:38_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 2</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E29">A4</font></span></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/></div><span class="kw">end;</span></div>
<a NAME="E2:38"/>
 <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 2 is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:38"/>
 <a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 2 <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> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" title="ASYMPT_1:func.3">seq_n^</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E26">A1</font></span></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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