<?xml version="1.0"?>
<div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c3">a1</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c4">b1</font> = <font color="Maroon" title="c2">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> </span>}</span> ;<br/>
<a NAME="E2:60_1_1"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">c</font> where <font color="Olive" title="b2">c</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b2">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> </span>}</span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> </span>}</span>  implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:60_1_1_1"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> </span>}</span> 
 ; <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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><br/>

<a NAME="E2:60_1_1_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
 ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:60_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></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">reconsider </span><font color="Maroon" title="c6">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E4:60_1_1"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 ex <font color="Olive" title="b1">d</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E40">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c7">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c8">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">d</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c7">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:60_1_1_2_1"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:60_1_1_2_1"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c7">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">b1</font> = <font color="Maroon" title="c8">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1;<br/><span class="kw">consider </span><font color="Maroon" title="c12">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:60_1_1_2_1"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T10" title="SEQ_4:th.10">SEQ_4:10</a></span>;<br/><a NAME="E6:60_1_1_2_1"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
<font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <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#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_2_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E7:60_1_1_2_1"/><span class="lab"><font color="Green" title="E45">A13</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"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1_2_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="prepower.html#T24" title="PREPOWER:th.24">PREPOWER:24</a></span>;<br/><a NAME="E8:60_1_1_2_1"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E9:60_1_1_2_1"/><span class="lab"><font color="Green" title="E47">A15</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"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:60_1_1_2_1"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E8:60_1_1_2_1"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:60_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A16</font></span>: 
<font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_2_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E11:60_1_1_2_1"/><span class="lab"><font color="Green" title="E49">A17</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <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#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_2_1"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E10:60_1_1_2_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E12:60_1_1_2_1"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1_1_2_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E11:60_1_1_2_1"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="xcmplx_1.html#T93" title="XCMPLX_1:th.93">XCMPLX_1:93</a></span>;<br/><a NAME="E13:60_1_1_2_1"/><span class="lab"><font color="Green" title="E51">A19</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_1_2_1"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E12:60_1_1_2_1"><span class="lab"><font color="Green" title="E50">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="E14:60_1_1_2_1"/><span class="lab"><font color="Green" title="E52">A20</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c8">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_2_1"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="txt" href="#E13:60_1_1_2_1"><span class="lab"><font color="Green" title="E51">A19</font></span></a>, <a class="ref" href="xreal_1.html#T87" title="XREAL_1:th.87">XREAL_1:87</a></span>;<br/><a NAME="E15:60_1_1_2_1"/><span class="lab"><font color="Green" title="E53">A21</font></span>: 
<font color="Maroon" title="c9">a1</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_2_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E14:60_1_1_2_1"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="ref" href="power.html#T33" target="_self" title="POWER:th.33">Th33</a></span>;<br/><span class="kw">thus </span><a NAME="E16:60_1_1_2_1"/>
 ex <font color="Olive" title="b1">d</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c7">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:60_1_1_2_1"><span class="lab"><font color="Green" title="E53">A21</font></span></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>
<div><span class="lab"><font color="Green" title="E41">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:60_1_1_2_2_1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:60_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E42">A23</font></span>: 
<font color="Maroon" title="c1">a</font> <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">  ex <font color="Olive" title="b1">d</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font></span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:60_1_1_2_2_1_1"/>
 ex <font color="Olive" title="b1">d</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E1:60_1_1_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E1:60_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E42">A23</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:60_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E42">A24</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font></span><br/><div class="add"><a NAME="E2:60_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E43">A25</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E1:60_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E42">A24</font></span></a>, <a class="ref" href="xreal_1.html#T90" title="XREAL_1:th.90">XREAL_1:90</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">d</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:60_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E44">A26</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E1:60_1_1_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E2:60_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E43">A25</font></span></a></span>;<br/><a NAME="E5:60_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E45">A27</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:60_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E44">A26</font></span></a>, <a class="ref" href="power.html#T37" target="_self" title="POWER:th.37">Th37</a></span>;<br/><span class="kw">thus </span><a NAME="E6:60_1_1_2_2_1_2"/>
 ex <font color="Olive" title="b1">e</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E45">A27</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">d</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:60_1_1_2"/><span class="lab"><font color="Green" title="E42">A28</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_2"><span class="lab"><font color="Green" title="E41">A22</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">d</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="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font></span><br/>

<a NAME="E5:60_1_1_2"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c7">d</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:60_1_1_2"/>
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1_2"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E5:60_1_1_2"><span class="lab"><font color="Green" title="E43">A29</font></span></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>
<div><span class="lab"><font color="Green" title="E40">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:60_1_1_3_1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:60_1_1_3_1_1"/><span class="lab"><font color="Green" title="E41">A31</font></span>: 
<font color="Maroon" title="c1">a</font> <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">  ex <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">d</font></span><br/><div class="add"><a NAME="E2:60_1_1_3_1_1"/><span class="lab"><font color="Green" title="E42">A32</font></span>: 
<font color="Maroon" title="c6">X</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c7">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:60_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E43">A33</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T10" title="SEQ_4:th.10">SEQ_4:10</a></span>;<br/>
<a NAME="E3:60_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E44">A34</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_3_1_1"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E4:60_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E45">A35</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="txt" href="#E3:60_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E44">A34</font></span></a>, <a class="ref" href="xreal_1.html#T79" title="XREAL_1:th.79">XREAL_1:79</a></span>;<br/>
<a NAME="E5:60_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E46">A36</font></span>: 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></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"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E45">A35</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E6:60_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E47">A37</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <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#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E7:60_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E48">A38</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"><font color="Maroon" title="c3">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="ref" href="prepower.html#T24" title="PREPOWER:th.24">PREPOWER:24</a></span>;<br/>
<a NAME="E8:60_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E49">A39</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E46">A36</font></span></a>, <a class="txt" href="#E7:60_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E48">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E50">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">c</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> implies <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:60_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A41</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</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="c8">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font></span><br/><a NAME="E2:60_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A42</font></span>: 
 ex <font color="Olive" title="b1">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E51">A41</font></span></a></span>;<br/><a NAME="E3:60_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A43</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:60_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E49">A39</font></span></a>, <a class="txt" href="#E2:60_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E52">A42</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">thus </span><a NAME="E4:60_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_3_1_1"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="txt" href="#E3:60_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="ref" href="power.html#T44" target="_self" title="POWER:th.44">Th44</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">thus </span><a NAME="E10:60_1_1_3_1_1_1"/>
<font color="Maroon" title="c6">X</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E50">A40</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</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">take </span><font color="Maroon" title="c7">d</font> =  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c6">X</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="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span><br/><a NAME="E3:60_1_1_3_1_1"/><span class="lab"><font color="Green" title="E43">A44</font></span>: 
not <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E44">A45</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">e</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E3:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E45">A46</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">e</font>
 <span class="kw">and </span><br/><a NAME="E4:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E46">A47</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">e</font> = <font color="Maroon" title="c8">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E47">A48</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T10" title="SEQ_4:th.10">SEQ_4:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">m</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> 1,<font color="Maroon" title="c10">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/>
<a NAME="E9:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E48">A49</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</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="E10:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E49">A50</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E47">A48</font></span></a>, <a class="txt" href="#E9:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E50">A51</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</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="txt" href="#E3:60"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E3:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="ref" href="xreal_1.html#T189" title="XREAL_1:th.189">XREAL_1:189</a></span>;<br/>
<a NAME="E12:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E51">A52</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E13:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E52">A53</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="txt" href="#E12:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E51">A52</font></span></a>, <a class="ref" href="xreal_1.html#T79" title="XREAL_1:th.79">XREAL_1:79</a></span>;<br/>
<a NAME="E14:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E53">A54</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</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="E15:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E54">A55</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="txt" href="#E14:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="ref" href="xreal_1.html#T85" title="XREAL_1:th.85">XREAL_1:85</a></span>;<br/>
<a NAME="E16:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E55">A56</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E17:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E56">A57</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <font color="Maroon" title="c3">a1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="power.html#T23" target="_self" title="POWER:th.23">Th23</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E18:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E57">A58</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E19:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E58">A59</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A56</font></span></a>, <a class="txt" href="#E18:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E20:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E59">A60</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E19:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A59</font></span></a>, <a class="ref" href="power.html#T52" target="_self" title="POWER:th.52">Th52</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E21:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E60">A61</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E20:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A60</font></span></a>, <a class="ref" href="xreal_1.html#T85" title="XREAL_1:th.85">XREAL_1:85</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">c</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E23:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E61">A62</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">and </span><br/><a NAME="E24:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E62">A63</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E2:60_1_1_3_1_1"><span class="lab"><font color="Green" title="E42">A32</font></span></a>, <a class="txt" href="#E14:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">c</font> = <font color="Maroon" title="c12">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E26:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E63">A64</font></span>: 
 ex <font color="Olive" title="b1">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E61">A62</font></span></a></span>;<br/>
<a NAME="E27:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E64">A65</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span> <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="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="power.html#T39" target="_self" title="POWER:th.39">Th39</a></span>;<br/>
<a NAME="E28:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E65">A66</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E63">A64</font></span></a>, <a class="txt" href="#E27:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E64">A65</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E29:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E66">A67</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A61</font></span></a>, <a class="txt" href="#E28:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E65">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E30:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E67">A68</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">c</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E29:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E66">A67</font></span></a>, <a class="ref" href="power.html#T32" target="_self" title="POWER:th.32">Th32</a></span>;<br/>
<a NAME="E31:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E68">A69</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">c</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E62">A63</font></span></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<a NAME="E32:60_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E69">A70</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">c</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_3_1_1"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="txt" href="#E31:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E68">A69</font></span></a>, <a class="ref" href="power.html#T44" target="_self" title="POWER:th.44">Th44</a></span>;<br/>
<span class="kw">thus </span><a NAME="E33:60_1_1_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E46">A47</font></span></a>, <a class="txt" href="#E30:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E67">A68</font></span></a>, <a class="txt" href="#E32:60_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E69">A70</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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><a NAME="E4:60_1_1_3_1_1"/><span class="lab"><font color="Green" title="E44">A71</font></span>: 
not <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E45">A72</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">e</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E3:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E46">A73</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">e</font>
 <span class="kw">and </span><br/><a NAME="E4:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E47">A74</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E45">A72</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">e</font> = <font color="Maroon" title="c8">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E6:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E48">A75</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c7">d</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="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="power.html#T39" target="_self" title="POWER:th.39">Th39</a></span>;<br/>
<a NAME="E7:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E49">A76</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</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="txt" href="#E3:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E46">A73</font></span></a>, <a class="txt" href="#E4:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E47">A74</font></span></a>, <a class="txt" href="#E6:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E48">A75</font></span></a>, <a class="ref" href="xreal_1.html#T189" title="XREAL_1:th.189">XREAL_1:189</a></span>;<br/>
<a NAME="E8:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E50">A77</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E49">A76</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  = $1 <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">s</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E10:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E51">A78</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></span>
<a NAME="E11:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E52">A79</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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> 1 holds <br/><font color="Maroon" title="c10">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E51">A78</font></span></a></span>;<br/>
<a NAME="E12:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E53">A80</font></span>: 
( <font color="Maroon" title="c10">s</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E11:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E52">A79</font></span></a>, <a class="ref" href="power.html#T24" target="_self" title="POWER:th.24">Th24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E14:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E54">A81</font></span>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">n</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</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="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A77</font></span></a>, <a class="txt" href="#E12:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E53">A80</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">m</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> 1,<font color="Maroon" title="c11">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/>
<a NAME="E16:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E55">A82</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</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="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E54">A81</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E17:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E56">A83</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</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="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E51">A78</font></span></a>, <a class="txt" href="#E16:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E55">A82</font></span></a></span>;<br/>
<a NAME="E18:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E57">A84</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a></span>;<br/>
<a NAME="E19:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E58">A85</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E56">A83</font></span></a>, <a class="txt" href="#E18:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E57">A84</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E20:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E59">A86</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E58">A85</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E21:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E60">A87</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E20:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E59">A86</font></span></a>, <a class="ref" href="power.html#T52" target="_self" title="POWER:th.52">Th52</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E22:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E61">A88</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span></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="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E48">A75</font></span></a>, <a class="txt" href="#E21:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E60">A87</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<a NAME="E23:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E62">A89</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span></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="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E22:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E61">A88</font></span></a>, <a class="ref" href="power.html#T32" target="_self" title="POWER:th.32">Th32</a></span>;<br/>
<a NAME="E24:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E63">A90</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E46">A73</font></span></a>, <a class="txt" href="#E6:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E48">A75</font></span></a>, <a class="ref" href="xreal_1.html#T191" title="XREAL_1:th.191">XREAL_1:191</a></span>;<br/>
<a NAME="E25:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E64">A91</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E24:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E63">A90</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<a NAME="E26:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E65">A92</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E62">A89</font></span></a>, <a class="txt" href="#E25:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E64">A91</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E27:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E66">A93</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E65">A92</font></span></a></span>;<br/>
<a NAME="E28:60_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E67">A94</font></span>: 
( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c7">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_3_1_1"><span class="lab"><font color="Green" title="E42">A32</font></span></a>, <a class="txt" href="#E27:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E66">A93</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<span class="kw">thus </span><a NAME="E29:60_1_1_3_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:60_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E67">A94</font></span></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</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">thus </span><a NAME="E5:60_1_1_3_1_1"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_3_1_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E4:60_1_1_3_1_1"><span class="lab"><font color="Green" title="E44">A71</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:60_1_1_3_1_2"/><span class="lab"><font color="Green" title="E41">A95</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">d</font></span><br/><div class="add"><a NAME="E2:60_1_1_3_1_2"/><span class="lab"><font color="Green" title="E42">A96</font></span>: 
<font color="Maroon" title="c6">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c7">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:60_1_1_3_1_2_1"/><span class="lab"><font color="Green" title="E43">A97</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">a1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T10" title="SEQ_4:th.10">SEQ_4:10</a></span>;<br/>
<a NAME="E3:60_1_1_3_1_2_1"/><span class="lab"><font color="Green" title="E44">A98</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E1:60_1_1_3_1_2"><span class="lab"><font color="Green" title="E41">A95</font></span></a>, <a class="ref" href="xreal_1.html#T90" title="XREAL_1:th.90">XREAL_1:90</a></span>;<br/>
<a NAME="E4:60_1_1_3_1_2_1"/><span class="lab"><font color="Green" title="E45">A99</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A98</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E5:60_1_1_3_1_2_1"/><span class="lab"><font color="Green" title="E46">A100</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A97</font></span></a>, <a class="txt" href="#E4:60_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A99</font></span></a>, <a class="ref" href="xreal_1.html#T79" title="XREAL_1:th.79">XREAL_1:79</a></span>;<br/>
<a NAME="E6:60_1_1_3_1_2_1"/><span class="lab"><font color="Green" title="E47">A101</font></span>: 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></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"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E46">A100</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E7:60_1_1_3_1_2_1"/><span class="lab"><font color="Green" title="E48">A102</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <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#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E8:60_1_1_3_1_2_1"/><span class="lab"><font color="Green" title="E49">A103</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">a1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">a1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:60_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E48">A102</font></span></a>, <a class="ref" href="prepower.html#T24" title="PREPOWER:th.24">PREPOWER:24</a></span>;<br/>
<a NAME="E9:60_1_1_3_1_2_1"/><span class="lab"><font color="Green" title="E50">A104</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E47">A101</font></span></a>, <a class="txt" href="#E8:60_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E49">A103</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:60_1_1_3_1_2_1"/><span class="lab"><font color="Green" title="E51">A105</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E9:60_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E50">A104</font></span></a>, <a class="ref" href="power.html#T37" target="_self" title="POWER:th.37">Th37</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E52">A106</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">c</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> implies <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c7">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:60_1_1_3_1_2_1_1"/><span class="lab"><font color="Green" title="E53">A107</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</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="c8">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c7">n</font></span><br/><a NAME="E2:60_1_1_3_1_2_1_1"/><span class="lab"><font color="Green" title="E54">A108</font></span>: 
 ex <font color="Olive" title="b1">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_3_1_2_1_1"><span class="lab"><font color="Green" title="E53">A107</font></span></a></span>;<br/><a NAME="E3:60_1_1_3_1_2_1_1"/><span class="lab"><font color="Green" title="E55">A109</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E51">A105</font></span></a>, <a class="txt" href="#E2:60_1_1_3_1_2_1_1"><span class="lab"><font color="Green" title="E54">A108</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">thus </span><a NAME="E4:60_1_1_3_1_2_1_1"/>
<font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E1:60_1_1_3_1_2"><span class="lab"><font color="Green" title="E41">A95</font></span></a>, <a class="txt" href="#E3:60_1_1_3_1_2_1_1"><span class="lab"><font color="Green" title="E55">A109</font></span></a>, <a class="ref" href="power.html#T45" target="_self" title="POWER:th.45">Th45</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">thus </span><a NAME="E12:60_1_1_3_1_2_1"/>
<font color="Maroon" title="c6">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:60_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E52">A106</font></span></a>, <a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</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">take </span><font color="Maroon" title="c7">d</font> =  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">X</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="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span><br/><a NAME="E3:60_1_1_3_1_2"/><span class="lab"><font color="Green" title="E43">A110</font></span>: 
not <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E44">A111</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">e</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E3:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E45">A112</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">e</font>
 <span class="kw">and </span><br/><a NAME="E4:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E46">A113</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E44">A111</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">e</font> = <font color="Maroon" title="c8">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E47">A114</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">a1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T10" title="SEQ_4:th.10">SEQ_4:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">m</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> 1,<font color="Maroon" title="c10">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/>
<a NAME="E9:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E48">A115</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</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="E10:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E49">A116</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E47">A114</font></span></a>, <a class="txt" href="#E9:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E48">A115</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E50">A117</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</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="txt" href="#E3:60"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E3:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E45">A112</font></span></a>, <a class="ref" href="xreal_1.html#T189" title="XREAL_1:th.189">XREAL_1:189</a></span>;<br/>
<a NAME="E12:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E51">A118</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E50">A117</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E13:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E52">A119</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E49">A116</font></span></a>, <a class="txt" href="#E12:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E51">A118</font></span></a>, <a class="ref" href="xreal_1.html#T79" title="XREAL_1:th.79">XREAL_1:79</a></span>;<br/>
<a NAME="E14:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E53">A120</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</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="E15:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E54">A121</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E52">A119</font></span></a>, <a class="txt" href="#E14:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E53">A120</font></span></a>, <a class="ref" href="xreal_1.html#T85" title="XREAL_1:th.85">XREAL_1:85</a></span>;<br/>
<a NAME="E16:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E55">A122</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E54">A121</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E17:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E56">A123</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">a1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">a1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="power.html#T23" target="_self" title="POWER:th.23">Th23</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E18:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E57">A124</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E56">A123</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E19:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E58">A125</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E55">A122</font></span></a>, <a class="txt" href="#E18:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E57">A124</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E20:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E59">A126</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">a1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E19:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E58">A125</font></span></a>, <a class="ref" href="power.html#T52" target="_self" title="POWER:th.52">Th52</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E21:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E60">A127</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E20:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E59">A126</font></span></a>, <a class="ref" href="xreal_1.html#T85" title="XREAL_1:th.85">XREAL_1:85</a></span>;<br/>
<a NAME="E22:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E61">A128</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E21:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E60">A127</font></span></a>, <a class="ref" href="power.html#T37" target="_self" title="POWER:th.37">Th37</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">c</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E24:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E62">A129</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">and </span><br/><a NAME="E25:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E63">A130</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E2:60_1_1_3_1_2"><span class="lab"><font color="Green" title="E42">A96</font></span></a>, <a class="txt" href="#E14:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E53">A120</font></span></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">c</font> = <font color="Maroon" title="c12">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E27:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E64">A131</font></span>: 
 ex <font color="Olive" title="b1">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E62">A129</font></span></a></span>;<br/>
<a NAME="E28:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E65">A132</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <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="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="power.html#T39" target="_self" title="POWER:th.39">Th39</a></span>;<br/>
<a NAME="E29:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E66">A133</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E64">A131</font></span></a>, <a class="txt" href="#E28:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E65">A132</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E30:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E67">A134</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E61">A128</font></span></a>, <a class="txt" href="#E29:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E66">A133</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E31:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E68">A135</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">c</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E30:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E67">A134</font></span></a>, <a class="ref" href="power.html#T32" target="_self" title="POWER:th.32">Th32</a></span>;<br/>
<a NAME="E32:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E69">A136</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E63">A130</font></span></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<a NAME="E33:60_1_1_3_1_2_2"/><span class="lab"><font color="Green" title="E70">A137</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E1:60_1_1_3_1_2"><span class="lab"><font color="Green" title="E41">A95</font></span></a>, <a class="txt" href="#E32:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E69">A136</font></span></a>, <a class="ref" href="power.html#T45" target="_self" title="POWER:th.45">Th45</a></span>;<br/>
<span class="kw">thus </span><a NAME="E34:60_1_1_3_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E46">A113</font></span></a>, <a class="txt" href="#E31:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E68">A135</font></span></a>, <a class="txt" href="#E33:60_1_1_3_1_2_2"><span class="lab"><font color="Green" title="E70">A137</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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><a NAME="E4:60_1_1_3_1_2"/><span class="lab"><font color="Green" title="E44">A138</font></span>: 
not <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E45">A139</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">e</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E3:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E46">A140</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">e</font>
 <span class="kw">and </span><br/><a NAME="E4:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E47">A141</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E45">A139</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">e</font> = <font color="Maroon" title="c8">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E6:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E48">A142</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c7">d</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="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="power.html#T39" target="_self" title="POWER:th.39">Th39</a></span>;<br/>
<a NAME="E7:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E49">A143</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</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="txt" href="#E3:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E46">A140</font></span></a>, <a class="txt" href="#E4:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E47">A141</font></span></a>, <a class="txt" href="#E6:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E48">A142</font></span></a>, <a class="ref" href="xreal_1.html#T189" title="XREAL_1:th.189">XREAL_1:189</a></span>;<br/>
<a NAME="E8:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E50">A144</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E49">A143</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> = $1 <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">s</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E10:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E51">A145</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></span>
<a NAME="E11:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E52">A146</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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> 1 holds <br/><font color="Maroon" title="c10">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">a1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E51">A145</font></span></a></span>;<br/>
<a NAME="E12:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E53">A147</font></span>: 
( <font color="Maroon" title="c10">s</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E11:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E52">A146</font></span></a>, <a class="ref" href="power.html#T24" target="_self" title="POWER:th.24">Th24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E14:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E54">A148</font></span>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">n</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</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="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E50">A144</font></span></a>, <a class="txt" href="#E12:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E53">A147</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">m</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> 1,<font color="Maroon" title="c11">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/>
<a NAME="E16:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E55">A149</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</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="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E54">A148</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E17:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E56">A150</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</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="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E51">A145</font></span></a>, <a class="txt" href="#E16:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E55">A149</font></span></a></span>;<br/>
<a NAME="E18:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E57">A151</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a></span>;<br/>
<a NAME="E19:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E58">A152</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E56">A150</font></span></a>, <a class="txt" href="#E18:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E57">A151</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E20:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E59">A153</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E58">A152</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E21:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E60">A154</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">a1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E20:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E59">A153</font></span></a>, <a class="ref" href="power.html#T52" target="_self" title="POWER:th.52">Th52</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E22:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E61">A155</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span></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="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E48">A142</font></span></a>, <a class="txt" href="#E21:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E60">A154</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<a NAME="E23:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E62">A156</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span></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">b1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E22:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E61">A155</font></span></a>, <a class="ref" href="power.html#T37" target="_self" title="POWER:th.37">Th37</a></span>;<br/>
<a NAME="E24:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E63">A157</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span></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="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E23:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E62">A156</font></span></a>, <a class="ref" href="power.html#T32" target="_self" title="POWER:th.32">Th32</a></span>;<br/>
<a NAME="E25:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E64">A158</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E46">A140</font></span></a>, <a class="txt" href="#E6:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E48">A142</font></span></a>, <a class="ref" href="xreal_1.html#T191" title="XREAL_1:th.191">XREAL_1:191</a></span>;<br/>
<a NAME="E26:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E65">A159</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E25:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E64">A158</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<a NAME="E27:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E66">A160</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E63">A157</font></span></a>, <a class="txt" href="#E26:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E65">A159</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E28:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E67">A161</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E66">A160</font></span></a></span>;<br/>
<a NAME="E29:60_1_1_3_1_2_3"/><span class="lab"><font color="Green" title="E68">A162</font></span>: 
( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c7">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_3_1_2"><span class="lab"><font color="Green" title="E42">A96</font></span></a>, <a class="txt" href="#E28:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E67">A161</font></span></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<span class="kw">thus </span><a NAME="E30:60_1_1_3_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:60_1_1_3_1_2_3"><span class="lab"><font color="Green" title="E68">A162</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</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">thus </span><a NAME="E5:60_1_1_3_1_2"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_3_1_2"><span class="lab"><font color="Green" title="E43">A110</font></span></a>, <a class="txt" href="#E4:60_1_1_3_1_2"><span class="lab"><font color="Green" title="E44">A138</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E6:60_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1"><span class="lab"><font color="Green" title="E40">A30</font></span></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>
