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<a NAME="E1:67"/>
 ex <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 <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="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="Olive" title="b2">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b2">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c1">N</font> = 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 10 as    <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>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c2">N</font> = <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c2">N</font> holds <br/><font color="Olive" title="b1">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b1">n</font></span>)</span></span><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="Red">::  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> <font color="Maroon" title="c2">N</font> implies <font color="Maroon" title="c3">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> )</span><br/><span class="kw">set </span><font color="Maroon" title="c4">x</font> =  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font>;<br/><a NAME="E1:67_1_1"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 9 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></span>;<br/><span class="kw">assume </span><a NAME="E2:67_1_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span><br/><a NAME="E3:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<font color="Maroon" title="c3">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></span>;<br/><a NAME="E4:67_1_1"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 10 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></span>;<br/><a NAME="E5:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:67_1_1"/><span class="kw">then </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="power.html#T29" title="POWER:th.29">POWER:29</a></span>;<br/><a NAME="E7:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/><a NAME="E8:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></span>;<br/><a NAME="E9:67_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <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 href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E10:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A4</font></span>: 
4 <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="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <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="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E11:67_1_1"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 10 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></span>;<br/><a NAME="E12:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></span>;<br/><a NAME="E14:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A6</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E15:67_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:67_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E16:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:67_1_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default">2 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/><a NAME="E18:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T53" title="POWER:th.53">POWER:53</a></span>;<br/><a NAME="E19:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/><a NAME="E20:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A7</font></span>: 
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></span>;<br/><a NAME="E21:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A8</font></span>: 
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></span>;<br/><a NAME="E22:67_1_1"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 10 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 9
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></span>;<br/><a NAME="E23:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 9
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E24:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 9</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/><a NAME="E25:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 9 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/><a NAME="E26:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A9</font></span>: 
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 9 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></span>;<br/><a NAME="E27:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A10</font></span>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <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 href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E28:67_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <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 href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E29:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:67_1_1"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="asympt_1.html#E61"><span class="lab"><font color="Green" title="E38">Lm31</font></span></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/><a NAME="E30:67_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/><a NAME="E31:67_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></span>;<br/><a NAME="E32:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T53" title="POWER:th.53">POWER:53</a></span>;<br/><a NAME="E33:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:67_1_1"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/><a NAME="E34:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 2</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/><a NAME="E35:67_1_1"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T53" title="POWER:th.53">POWER:53</a></span>;<br/><a NAME="E36:67_1_1"/><span class="kw">then </span>
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></span>;<br/><a NAME="E37:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="power.html#D3" title="POWER:def.3">POWER:def 3</a></span>;<br/><a NAME="E38:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:67_1_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/><a NAME="E39:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T53" title="POWER:th.53">POWER:53</a></span>;<br/><a NAME="E40:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:67_1_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="power.html#D3" title="POWER:def.3">POWER:def 3</a></span>;<br/><a NAME="E41:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 2</span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T53" title="POWER:th.53">POWER:53</a></span>;<br/><a NAME="E42:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/><a NAME="E43:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <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="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_1_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E44:67_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T83" title="XREAL_1:th.83">XREAL_1:83</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:67_1"/>
 ex <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 <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="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="Olive" title="b2">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c1">N3</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> <span class="kw"> such that </span><br/><a NAME="E3:67"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
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="c1">N3</font> holds <br/><font color="Olive" title="b1">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b1">n</font></span>)</span>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c2">N</font> = 30; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c2">N</font> holds <br/><font color="Olive" title="b1">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b1">n</font></span>)</span></span><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="Red">::  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> <font color="Maroon" title="c2">N</font> implies <font color="Maroon" title="c3">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:67_2"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span><br/><a NAME="E2:67_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A13</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></span>;<br/><a NAME="E3:67_2"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="asympt_1.html#E60"><span class="lab"><font color="Green" title="E37">Lm30</font></span></a></span>;<br/><a NAME="E4:67_2"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_2"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/><a NAME="E5:67_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/><a NAME="E6:67_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 6</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></span>;<br/><a NAME="E7:67_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/><a NAME="E8:67_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E9:67_2"/>
<font color="Maroon" title="c3">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T83" title="XREAL_1:th.83">XREAL_1:83</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">N2</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> <span class="kw"> such that </span><br/><a NAME="E6:67"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
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="c2">N2</font> holds <br/><font color="Olive" title="b1">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b1">n</font></span>)</span>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c3">N</font> = 7; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="Olive" title="b1">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 6 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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="c4">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 6 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:67_3"/>
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 6 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1</span><br/><a NAME="E2:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 6
 <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="E3:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 6 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 6 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 6
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/><span class="kw">hence </span><a NAME="E4:67_3"/>
<font color="Maroon" title="c4">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 6 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">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> <span class="kw"> such that </span><br/><a NAME="E9:67"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
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">N1</font> holds <br/><font color="Olive" title="b1">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 6 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">N</font> =  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font></span>)</span>;<br/>
<a NAME="E10:67"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E11:67"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E12:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A17</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:67"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E14:67"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">N3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E15:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A19</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">N3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:67"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E16:67"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E17:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A20</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:67"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E18:67"/>
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E19:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A21</font></span>: 
 <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font></span>)</span> implies <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )</span><br/><a NAME="E1:67_4"/><span class="lab"><font color="Green" title="E49">A22</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span></span>)</span></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="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 ;<br/><span class="kw">assume </span><a NAME="E2:67_4"/><span class="lab"><font color="Green" title="E50">A23</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c3">N1</font>,2</span>)</span>,<span class="p1">(<span class="default"><a href="asympt_0.html#K4" title="ASYMPT_0:func.4">max</a> <font color="Maroon" title="c2">N2</font>,<font color="Maroon" title="c1">N3</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1</span><br/><a NAME="E3:67_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:67"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:67_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A24</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E41">A14</font></span></a></span>;<br/><a NAME="E5:67_4"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E2:67_4"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:67_4"/><span class="kw">then </span>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></span>;<br/><a NAME="E7:67_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A25</font></span>: 
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></span>;<br/><a NAME="E8:67_4"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:67"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E2:67_4"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:67_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 6 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E42">A15</font></span></a></span>;<br/><a NAME="E10:67_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 6</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_4"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E11:67_4"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">N3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:67"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E2:67_4"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:67_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A27</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E40">A11</font></span></a></span>;<br/><a NAME="E13:67_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 6</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 ;<br/><a NAME="E14:67_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_4"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E12:67_4"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E1:67_4"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E15:67_4"/><span class="kw">then </span>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T95" title="SQUARE_1:th.95">SQUARE_1:95</a>, <a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/><a NAME="E16:67_4"/><span class="kw">then </span>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67_4"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>;<br/><span class="kw">hence </span><a NAME="E17:67_4"/>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Maroon" title="c5">n</font></span>)</span> <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="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E21:67"/>
 ex <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 <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="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/><span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<font color="Olive" title="b2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
