<?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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</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="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 ;<br/>

<i><font color="Green">E34</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T12">NEWTON:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">m</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
;<br/>
<a NAME="E4:34"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="real_2.html#T138">REAL_2:138</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_2"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">a</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="34_2_1"><b>suppose </b></a><a NAME="E1:34_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:34_2_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:34_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</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="newton.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E4:34_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="txt" href="prepower.html#E1:34"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E5:34_2_1"/>
<font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="txt" href="prepower.html#E1:34"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E6:34_2_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E3:34"><i><font color="Green">E34</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E7:34_2_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</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="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</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#T2" target="_self">Th2</a>, <a class="txt" href="prepower.html#E2:34"><i><font color="Green">E32</font></i></a>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</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="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</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="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1"><b>suppose </b></a><a NAME="E1:34_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</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="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="prepower.html#T2" target="_self">Th2</a>, <a class="txt" href="prepower.html#E2:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="prepower.html#E4:34"><i><font color="Green">E35</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_2"><b>suppose </b></a><a NAME="E1:34_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</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="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="prepower.html#T2" target="_self">Th2</a>, , <a class="txt" href="prepower.html#E1:34_2_1"><i><font color="Green">E36</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_2"><b>suppose </b></a><a NAME="E1:34_2_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">n</font>, <font color="Maroon">k1</font> = 1 as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><i><font color="Green">E62</font></i>: <font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> = 
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="prepower.html#E1:34"><i><font color="Green">E31</font></i></a></i>
;<br/><a NAME="E5:34_2_2"/>
<font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="prepower.html#E1:34"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:34_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="prepower.html#T2" target="_self">Th2</a>, <a class="txt" href="prepower.html#E2:34"><i><font color="Green">E32</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
