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<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> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 ;<br/>

<b>hence </b><a NAME="E2:24"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</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="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</a></i>;<br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green" title="E20">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="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_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="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> or <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="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">a</font> )
 </span><b>by </b><i><a class="txt" href="prepower.html#E3:24"><i><font color="Green" title="E20">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="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/><i><font color="Green" title="E21">A3</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="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="prepower.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="prepower.html#T13" target="_self" title="PREPOWER:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="prepower.html#E1:24_1_1"><i><font color="Green" title="E21">A3</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="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/><i><font color="Green" title="E21">A4</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="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">a</font>
 ;<br/><div class="add"><a NAME="E2:24_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="prepower.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="prepower.html#T13" target="_self" title="PREPOWER:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E3:24_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="prepower.html#E1:24_1_2"><i><font color="Green" title="E21">A4</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>

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