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

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="rat_1.html#NM1">Rational</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>





<b>assume </b><a NAME="E1:74"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:74"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:74"/><i><font color="Green">E34</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="rat_1.html#T29">RAT_1:29</a></i>;<br/>
<a NAME="E4:74"/><i><font color="Green">E35</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="rat_1.html#T29">RAT_1:29</a></i>;<br/>
<a NAME="E5:74"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E11:74"><i><font color="Green">E60</font></i></a>, </i>;<br/>
<a NAME="E6:74"/><i><font color="Green">E43</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rat_1.html#D3">RAT_1:def 3</a></i>;<br/>
<a NAME="E7:74"/>
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rat_1.html#D3">RAT_1:def 3</a></i>;<br/>
<a NAME="E8:74"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T26" target="_self">Th26</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E9:74"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E10:74"/><i><font color="Green">E59</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rat_1.html#T27">RAT_1:27</a></i>;<br/>
<a NAME="E11:74"/><i><font color="Green">E60</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rat_1.html#D3">RAT_1:def 3</a></i>;<br/>
<a NAME="E12:74"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rat_1.html#T37">RAT_1:37</a></i>;<br/>
<b>then </b><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rat_1.html#T37">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">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T84">XCMPLX_1:84</a></i>
;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E15:74"/><i><font color="Green">E61</font></i>: 
( <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="rat_1.html#T60">RAT_1:60</a></i>;<br/>
<a NAME="E16:74"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="prepower.html#D3" target="_self">Def3</a>, , <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<a NAME="E17:74"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E18:74"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E19:74"/><i><font color="Green">E63</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="rat_1.html#T29">RAT_1:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">n</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E21:74"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E11:74"><i><font color="Green">E60</font></i></a>, </i>;<br/>
<a NAME="E22:74"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E23:74"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E11:74"><i><font color="Green">E60</font></i></a>, </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E15:74"><i><font color="Green">E61</font></i></a>, <a class="txt" href="prepower.html#E18:74"><i><font color="Green">E62</font></i></a>, <a class="ref" href="prepower.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K2">numerator</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E18:74"><i><font color="Green">E62</font></i></a>, <a class="txt" href="prepower.html#E19:74"><i><font color="Green">E63</font></i></a>, <a class="txt" href="prepower.html#E21:74"><i><font color="Green">E64</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2">numerator</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K6" target="_self">#Z</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T12" target="_self">Th12</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">denominator</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2">numerator</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K6" target="_self">#Z</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="prepower.html#E23:74"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="prepower.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1">denominator</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2">numerator</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="prepower.html#E5:74"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="txt" href="prepower.html#E22:74"><i><font color="Green">E72</font></i></a>, <a class="txt" href="prepower.html#E23:74"><i><font color="Green">E73</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i>, , </i>
;<br/>


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