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

<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E26</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E3:6"/><i><font color="Green">E28</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:6"/><i><font color="Green">E29</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">pp</font> = 1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>, <font color="Maroon">qq</font> = 1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E6:6"/><i><font color="Green">E30</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:6"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">qq</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">pp</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:6"/>
1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E9:6"/><b>then </b>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">pp</font>
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E10:6"/><b>then </b><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">qq</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:6"/><b>then </b><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T150">REAL_2:150</a></i>;<br/>
<a NAME="E12:6"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:6"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E14:6"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E15:6"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1"/><i><font color="Green">E45</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:6_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> or 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1"><b>case </b></a><a NAME="E1:6_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:6_1_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E3:6_1_1_1_1"/><b>then </b>
0 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:6_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2"><b>case </b></a><a NAME="E1:6_1_1_1_2"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:6_1_1_1_2"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><b>hence </b><a NAME="E3:6_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:6"/><i><font color="Green">E47</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<a NAME="E1:6_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_2_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> or 0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_1"><b>case </b></a><a NAME="E1:6_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:6_2_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E3:6_2_1_1_1"/><b>then </b>
0 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:6_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_2"><b>case </b></a><a NAME="E1:6_2_1_1_2"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:6_2_1_1_2"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><b>hence </b><a NAME="E3:6_2_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E17:6"/><i><font color="Green">E49</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_3_1"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_1"><b>case </b></a><a NAME="E1:6_3_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:6_3_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:6_3_1_1"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_3_1_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_3_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:6_3_1_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>thus </b><a NAME="E2:6_3_1_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_3_1_1_1_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 ;<br/><a NAME="E2:6_3_1_1_1_1_1_1_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><a NAME="E3:6_3_1_1_1_1_1_1_1_1"/><b>then </b>
0 <a href="real_1.html#K4">*</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E4:6_3_1_1_1_1_1_1_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E5:6_3_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><a NAME="E6:6_3_1_1_1_1_1_1_1_1"/><i><font color="Green">E57</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:6_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="power.html#T38">POWER:38</a></i>;<br/>
<a NAME="E3:6_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> 1
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>;<br/>
<a NAME="E4:6_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<b>hence </b><a NAME="E5:6_3_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> = 
0
<b>by </b><i>, , <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font>
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a>, </i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_3_1_1_1_1_1_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font>
 ;<br/><i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> = 
0 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>
<b>by </b><i>, , <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
0

;<br/><i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font> = 
0 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font>
<b>by </b><i>, , , <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
0

;<br/><b>thus </b><a NAME="E4:6_3_1_1_1_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_3_1_1_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:6_3_1_1_1_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>thus </b><a NAME="E2:6_3_1_1_1_1_1_2"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_3_1_1_1_1_1_2_1"/><i><font color="Green">E63</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_3_1_1_1_1_1_2_1_1"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 ;<br/><a NAME="E2:6_3_1_1_1_1_1_2_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><a NAME="E3:6_3_1_1_1_1_1_2_1_1"/><b>then </b>
0 <a href="real_1.html#K4">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E4:6_3_1_1_1_1_1_2_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E5:6_3_1_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><a NAME="E6:6_3_1_1_1_1_1_2_1_1"/><i><font color="Green">E67</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_3_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:6_3_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:6_3_1_1_1_1_1_2_1_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#T38">POWER:38</a></i>;<br/>
<a NAME="E4:6_3_1_1_1_1_1_2_1_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> 1
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>;<br/>
<b>hence </b><a NAME="E5:6_3_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="holder_1.html#E1:6"><i><font color="Green">E26</font></i></a>, <a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> = 
0
<b>by </b><i>, , <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a>, </i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_3_1_1_1_1_1_2_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font>
 ;<br/><i><font color="Green">E71</font></i>: <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font> = 
0 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font>
<b>by </b><i>, , <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
0

;<br/><i><font color="Green">E72</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> = 
0 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>
<b>by </b><i>, , , <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
0

;<br/><b>thus </b><a NAME="E4:6_3_1_1_1_1_1_2_1_2"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="holder_1.html#E1:6"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_3_1_1_1_1_1_2_1"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_3_1_1_1"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> )
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:6_3_1_1"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> &amp; ( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> implies <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> ) &amp; ( <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> implies <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_2"><b>case </b></a><a NAME="E1:6_3_1_2"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:6_3_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E3:6_3_1_2"/><i><font color="Green">E74</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:6_3_1_2"/><i><font color="Green">E75</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:6_3_1_2"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E6:6_3_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><a NAME="E7:6_3_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><a NAME="E8:6_3_1_2"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font> )
 <b>by </b><i>, , , , </i>;<br/><a NAME="E9:6_3_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><a NAME="E10:6_3_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><b>hence </b><a NAME="E11:6_3_1_2"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> &amp; ( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> implies <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> ) &amp; ( <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> implies <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E19:6"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> &amp; ( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> implies <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> ) &amp; ( <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font> implies <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> ) )
 ;<br/>


</div>
