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<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">x</font> holds <br/><font color="Olive" title="b2">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Olive" title="b2">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">N0</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:3"/><span class="lab"><font color="Green" title="E4">P1</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c2">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Olive" title="b1">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T2" target="_self" onmouseover="rs('asympt_3/T2')" onmouseout="rh()">LMC31</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">N</font> = <font color="Maroon" title="c2">N0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">N0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c2">N0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Olive" title="b1">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Olive" title="b1">x</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c2">N0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Olive" title="b1">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">N0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> implies <font color="Maroon" title="c4">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_1"/><span class="lab"><font color="Green" title="E5">AS2</font></span>: 
<font color="Maroon" title="c2">N0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">x</font></span><br/><a NAME="E2:3_1"/>
<font color="Maroon" title="c2">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">N0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/><a NAME="E3:3_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">N0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E5">AS2</font></span></a></span>;<br/><a NAME="E4:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">E1</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Maroon" title="c4">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E4">P1</font></span></a></span>;<br/><a NAME="E5:3_1"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">N0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E6:3_1"/><span class="kw">then </span>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E5">AS2</font></span></a></span>;<br/><a NAME="E7:3_1"/><span class="kw">then </span>
 <a href="power.html#K3" title="POWER:func.3">log</a> (2,2) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Maroon" title="c4">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_ff.html#T10" onmouseover="rs('pre_ff/T10')" onmouseout="rh()">PRE_FF:10</a></span>;<br/><a NAME="E8:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">P3</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Maroon" title="c4">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T52" onmouseover="rs('power/T52')" onmouseout="rh()">POWER:52</a></span>;<br/><a NAME="E9:3_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Maroon" title="c4">x</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="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Maroon" title="c4">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a>, <a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E6">E1</font></span></a></span>;<br/><a NAME="E10:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">P4</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:3_1"><span class="lab"><font color="Green" title="E7">P3</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">k</font></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T33" onmouseover="rs('power/T33')" onmouseout="rh()">POWER:33</a></span>
<br/>.= 
<font color="Maroon" title="c4">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E5">AS2</font></span></a>, <a class="ref" href="power.html#D3" onmouseover="rs('power/D3')" onmouseout="rh()">POWER:def 3</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:3_1"/>
<font color="Maroon" title="c4">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:3_1"><span class="lab"><font color="Green" title="E8">P4</font></span></a>, <a class="ref" href="power.html#T39" onmouseover="rs('power/T39')" onmouseout="rh()">POWER:39</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E4:3"/>
 for <font color="Olive" title="b1">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c2">N0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Olive" title="b1">x</font> <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Olive" title="b1">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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