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

<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <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="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> implies ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <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="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp; ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> implies <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> implies <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> ) ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:6"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">and </span><br/><a NAME="E2:6"/><span class="lab"><font color="Green" title="E5">A2</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">p</font></span>)</span> <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="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E3:6"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E4:6"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <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="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp; ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> implies <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> implies <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> ) )</span><br/>

<a NAME="E5:6"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_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="E1:6_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E6">A3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:6_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span><br/><div class="add"><a NAME="E2:6_1_1_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><span class="kw">hence </span><a NAME="E3:6_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:6_1_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span><br/><div class="add"><a NAME="E2:6_1_1_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:6_1_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:6_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c5">pp</font> = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E7:6"/>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</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="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="xreal_1.html#T189" title="XREAL_1:th.189">XREAL_1:189</a></span>;<br/>
<a NAME="E8:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A6</font></span>: 
1 <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> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">pp</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E9:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A7</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/>
<a NAME="E10:6"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_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="E1:6_2_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E7">A4</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:6_2_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span><br/><div class="add"><a NAME="E2:6_2_1_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><span class="kw">hence </span><a NAME="E3:6_2_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E10">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:6_2_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span><br/><div class="add"><a NAME="E2:6_2_1_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:6_2_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:6_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_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="E1:6_3_1"/>
( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_1"><span class="kw">case </span></a><a NAME="E1:6_3_1_1"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <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="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp; ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> implies <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> implies <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> ) )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_1_1_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="E1:6_3_1_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_3_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:6_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> iff <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E14">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:6_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span><br/><a NAME="E2:6_3_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E1:6_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>;<br/><a NAME="E3:6_3_1_1_1_1_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">q</font>
 ;<br/><a NAME="E4:6_3_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/><a NAME="E5:6_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A12</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</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="c2">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E8:6"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>;<br/><a NAME="E6:6_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E16">A13</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:6_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:6_3_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E5:6_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="power.html#T33" title="POWER:th.33">POWER:33</a></span>;<br/>
<a NAME="E3:6_3_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="xcmplx_1.html#T106" title="XCMPLX_1:th.106">XCMPLX_1:106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:6_3_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E1:6_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E6:6_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:6_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span></span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A15</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E1:6_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E1:6_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">hence </span><a NAME="E4:6_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_3_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E2:6_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:6_3_1_1_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> iff <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:6_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A16</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> iff <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> )</span><br/><div class="add"><a NAME="E2:6_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A17</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</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="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/><div><span class="lab"><font color="Green" title="E15">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:6_3_1_1_1_1_2_1"/>
<font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span><br/><a NAME="E2:6_3_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E1:6_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>;<br/><a NAME="E3:6_3_1_1_1_1_2_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">p</font>
 ;<br/><a NAME="E4:6_3_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/><a NAME="E5:6_3_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A19</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</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="c1">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>;<br/><a NAME="E6:6_3_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E17">A20</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:6_3_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E18">A21</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:6_3_1_1_1_1_2_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E5:6_3_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E16">A19</font></span></a>, <a class="ref" href="power.html#T33" title="POWER:th.33">POWER:33</a></span>;<br/>
<a NAME="E3:6_3_1_1_1_1_2_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T106" title="XCMPLX_1:th.106">XCMPLX_1:106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:6_3_1_1_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_3_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E18">A21</font></span></a>, <a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E1:6_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E6:6_3_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E17">A20</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:6_3_1_1_1_1_2_2"/>
<font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span></span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A22</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E1:6_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E1:6_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">hence </span><a NAME="E4:6_3_1_1_1_1_2_2"/>
<font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_3_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E2:6_3_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E16">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:6_3_1_1_1_1_2"/>
( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> iff <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:6_3_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <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="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp; ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> implies <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> implies <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E10:6"><span class="lab"><font color="Green" title="E11">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_2"><span class="kw">case </span></a><a NAME="E1:6_3_1_2"/><span class="lab"><font color="Green" title="E12">A23</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <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="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp; ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> implies <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> implies <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> ) )</span><br/><div class="add"><a NAME="E2:6_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A24</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:6_3_1_2"/><span class="lab"><font color="Green" title="E14">A25</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E1:6_3_1_2"><span class="lab"><font color="Green" title="E12">A23</font></span></a></span>;<br/><a NAME="E4:6_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> iff <font color="Maroon" title="c3">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E2:6_3_1_2"><span class="lab"><font color="Green" title="E13">A24</font></span></a>, <a class="ref" href="holder_1.html#T4" target="_self" title="HOLDER_1:th.4">Th4</a></span>;<br/><a NAME="E5:6_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A26</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> iff <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E2:6_3_1_2"><span class="lab"><font color="Green" title="E13">A24</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>;<br/><a NAME="E6:6_3_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <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="c3">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E2:6_3_1_2"><span class="lab"><font color="Green" title="E13">A24</font></span></a>, <a class="txt" href="#E3:6_3_1_2"><span class="lab"><font color="Green" title="E14">A25</font></span></a>, <a class="ref" href="holder_1.html#T4" target="_self" title="HOLDER_1:th.4">Th4</a></span>;<br/><a NAME="E7:6_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <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="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E2:6_3_1_2"><span class="lab"><font color="Green" title="E13">A24</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:6_3_1_2"/>
( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <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="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp; ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> implies <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> implies <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E3:6_3_1_2"><span class="lab"><font color="Green" title="E14">A25</font></span></a>, <a class="txt" href="#E5:6_3_1_2"><span class="lab"><font color="Green" title="E15">A26</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:6"/>
( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <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="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp; ( <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> implies <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> implies <font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
