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

<b>let </b><font color="Maroon" title="c1">a</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>  st <font color="Maroon" title="c1">a</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>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Olive" title="b1">p</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="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">q</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c2">p</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="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:67"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c1">a</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c2">p</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="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c4">dp</font> =  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font>;<br/>
<b>set </b><font color="Maroon" title="c5">dq</font> =  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font>;<br/>
<b>set </b><font color="Maroon" title="c6">np</font> =  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font>;<br/>
<b>set </b><font color="Maroon" title="c7">nq</font> =  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font>;<br/>
<a NAME="E2:67"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prepower.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="prepower.html#T49" target="_self" title="PREPOWER:th.49">Th49</a></i>;<br/>
<a NAME="E3:67"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prepower.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="prepower.html#T49" target="_self" title="PREPOWER:th.49">Th49</a></i>;<br/>
<a NAME="E4:67"/><i><font color="Green" title="E57">A4</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</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>  )
 
<b>by </b><i><a class="txt" href="prepower.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="prepower.html#T49" target="_self" title="PREPOWER:th.49">Th49</a></i>;<br/>
<a NAME="E5:67"/><i><font color="Green" title="E58">A5</font></i>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="rat_1.html#T29" title="RAT_1:th.29">RAT_1:29</a></i>;<br/>
<a NAME="E6:67"/><i><font color="Green" title="E59">A6</font></i>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="rat_1.html#T29" title="RAT_1:th.29">RAT_1:29</a></i>;<br/>
<a NAME="E7:67"/><i><font color="Green" title="E60">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</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> 
 
<b>by </b><i><a class="txt" href="prepower.html#E3:67"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="prepower.html#T13" target="_self" title="PREPOWER:th.13">Th13</a></i>;<br/>
<a NAME="E8:67"/><i><font color="Green" title="E61">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</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> 
 
<b>by </b><i><a class="txt" href="prepower.html#E2:67"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="prepower.html#T13" target="_self" title="PREPOWER:th.13">Th13</a></i>;<br/>
<a NAME="E9:67"/><i><font color="Green" title="E62">A9</font></i>: 
<span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="prepower.html#E5:67"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="prepower.html#E6:67"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="real_2.html#T138" title="REAL_2:th.138">REAL_2:138</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">ddp</font> =  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">ddq</font> =  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E12:67"/><i><font color="Green" title="E63">A10</font></i>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rat_1.html#D3" title="RAT_1:def.3">RAT_1:def 3</a></i>;<br/>
<a NAME="E13:67"/><i><font color="Green" title="E64">A11</font></i>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rat_1.html#D3" title="RAT_1:def.3">RAT_1:def 3</a></i>;<br/>
<a NAME="E14:67"/><b>then </b><i><font color="Green" title="E65">A12</font></i>: 
<span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prepower.html#E12:67"><i><font color="Green" title="E63">A10</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/>
<font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">q</font>
<b>by </b><i><a class="ref" href="rat_1.html#T37" title="RAT_1:th.37">RAT_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rat_1.html#T37" title="RAT_1:th.37">RAT_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></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"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E12:67"><i><font color="Green" title="E63">A10</font></i></a>, <a class="txt" href="prepower.html#E13:67"><i><font color="Green" title="E64">A11</font></i></a>, <a class="ref" href="xcmplx_1.html#T117" title="XCMPLX_1:th.117">XCMPLX_1:117</a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c10">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E17:67"/><i><font color="Green" title="E66">A13</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">n</font> &amp; <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">n</font> )
 <b>by </b><i><a class="txt" href="prepower.html#E14:67"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="rat_1.html#T60" title="RAT_1:th.60">RAT_1:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">k</font> = <font color="Maroon" title="c10">n</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E19:67"/><i><font color="Green" title="E67">A14</font></i>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rat_1.html#T27" title="RAT_1:th.27">RAT_1:27</a></i>;<br/>
<a NAME="E20:67"/><i><font color="Green" title="E68">A15</font></i>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rat_1.html#T27" title="RAT_1:th.27">RAT_1:27</a></i>;<br/>
<a NAME="E21:67"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prepower.html#E9:67"><i><font color="Green" title="E62">A9</font></i></a>, <a class="txt" href="prepower.html#E17:67"><i><font color="Green" title="E66">A13</font></i></a>, <a class="txt" href="prepower.html#E19:67"><i><font color="Green" title="E67">A14</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/>
<a NAME="E22:67"/><b>then </b>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prepower.html#E20:67"><i><font color="Green" title="E68">A15</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E23:67"/><b>then </b><i><font color="Green" title="E69">A16</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E24:67"/><i><font color="Green" title="E70">A17</font></i>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="rat_1.html#T29" title="RAT_1:th.29">RAT_1:29</a></i>;<br/>
<a NAME="E25:67"/><i><font color="Green" title="E71">A18</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prepower.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="prepower.html#T49" target="_self" title="PREPOWER:th.49">Th49</a></i>;<br/>
<a NAME="E26:67"/><i><font color="Green" title="E72">A19</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prepower.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="prepower.html#T63" target="_self" title="PREPOWER:th.63">Th63</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c2">p</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="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c3">q</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="prepower.html#T54" target="_self" title="PREPOWER:th.54">Th54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E2:67"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="prepower.html#E4:67"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="prepower.html#T31" target="_self" title="PREPOWER:th.31">Th31</a>, <a class="ref" href="rat_1.html#T29" title="RAT_1:th.29">RAT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E2:67"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="prepower.html#E5:67"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="prepower.html#E6:67"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="prepower.html#T35" target="_self" title="PREPOWER:th.35">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p0">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E3:67"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="prepower.html#E6:67"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="prepower.html#E31"><i><font color="Green" title="E18">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E5:67"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="prepower.html#E6:67"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="prepower.html#E7:67"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="prepower.html#T34" target="_self" title="PREPOWER:th.34">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E7:67"><i><font color="Green" title="E60">A7</font></i></a>, <a class="txt" href="prepower.html#E8:67"><i><font color="Green" title="E61">A8</font></i></a>, <a class="txt" href="prepower.html#E9:67"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="prepower.html#T31" target="_self" title="PREPOWER:th.31">Th31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">ddp</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">ddq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T46" target="_self" title="PREPOWER:th.46">Th46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">ddp</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">ddq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <font color="Maroon" title="c8">ddp</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T46" target="_self" title="PREPOWER:th.46">Th46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">ddp</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">ddq</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <font color="Maroon" title="c8">ddp</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T55" target="_self" title="PREPOWER:th.55">Th55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">ddp</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">ddq</font></span>)</span></span>)</span></span>)</span> <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="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">ddp</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T55" target="_self" title="PREPOWER:th.55">Th55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">ddp</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">ddq</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">ddp</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="prepower.html#T54" target="_self" title="PREPOWER:th.54">Th54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <font color="Maroon" title="c11">k</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E17:67"><i><font color="Green" title="E66">A13</font></i></a>, <a class="ref" href="prepower.html#T55" target="_self" title="PREPOWER:th.55">Th55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="txt" href="prepower.html#E9:67"><i><font color="Green" title="E62">A9</font></i></a>, <a class="txt" href="prepower.html#E17:67"><i><font color="Green" title="E66">A13</font></i></a>, <a class="txt" href="prepower.html#E25:67"><i><font color="Green" title="E71">A18</font></i></a>, <a class="ref" href="prepower.html#T56" target="_self" title="PREPOWER:th.56">Th56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">n</font>
<b>by </b><i><a class="ref" href="prepower.html#T46" target="_self" title="PREPOWER:th.46">Th46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">n</font>
<b>by </b><i><a class="txt" href="prepower.html#E23:67"><i><font color="Green" title="E69">A16</font></i></a>, <a class="txt" href="prepower.html#E24:67"><i><font color="Green" title="E70">A17</font></i></a>, <a class="txt" href="prepower.html#E25:67"><i><font color="Green" title="E71">A18</font></i></a>, <a class="ref" href="prepower.html#T34" target="_self" title="PREPOWER:th.34">Th34</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E23:67"><i><font color="Green" title="E69">A16</font></i></a>, <a class="txt" href="prepower.html#E26:67"><i><font color="Green" title="E72">A19</font></i></a>, <a class="txt" href="prepower.html#E31"><i><font color="Green" title="E18">Lm2</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
