<?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> ;<br/><b>let </b><font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">m</font> be    <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> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c1">a</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>  &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1"/>
( <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>  or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="prepower.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>suppose </b></a><a NAME="E1:31_1_1"/><i><font color="Green" title="E27">A3</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> 
 ;<br/><div class="add"><a NAME="E2:31_1_1"/><b>then </b><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <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> 
 <b>by </b><i><a class="txt" href="prepower.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</a></i>;<br/><i><font color="Green" title="E29">A5</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="txt" href="prepower.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="prepower.html#E2:31_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="prepower.html#E31"><i><font color="Green" title="E18">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="txt" href="prepower.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="prepower.html#T28" target="_self" title="PREPOWER:th.28">Th28</a></i>
;<br/><a NAME="E4:31_1_1"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="real_2.html#T138" title="REAL_2:th.138">REAL_2:138</a></i>;<br/><a NAME="E5:31_1_1"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</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#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="prepower.html#E2:31_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</a></i>;<br/><a NAME="E6:31_1_1"/><i><font color="Green" title="E32">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <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> 
 <b>by </b><i><a class="txt" href="prepower.html#E1:31_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="prepower.html#E4:31_1_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_2"/>
( <font color="Maroon" title="c2">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="prepower.html#E2:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2_1"><b>suppose </b></a><a NAME="E1:31_1_1_2_1"/>
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E1:31_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="prepower.html#E3:31_1_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="prepower.html#E4:31_1_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="prepower.html#E5:31_1_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2_2"><b>suppose </b></a><a NAME="E1:31_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E1:31_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="prepower.html#E3:31_1_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="prepower.html#E4:31_1_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="prepower.html#E6:31_1_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</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="31_1_2"><b>suppose </b></a><a NAME="E1:31_1_2"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:31_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="real_2.html#T138" title="REAL_2:th.138">REAL_2:138</a></i>;<br/><a NAME="E3:31_1_2"/><b>then </b><i><font color="Green" title="E28">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="prepower.html#E1:31_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</a></i>;<br/><a NAME="E4:31_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="prepower.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="prepower.html#E1:31_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:31_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="prepower.html#E2:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="prepower.html#E3:31_1_2"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
