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<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>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green">E31</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1"/>
( <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><a class="txt" href="prepower.html#E1:38"><i><font color="Green">E31</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_1_1_1"/>
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_1_1_2"/>
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E2:38"><i><font color="Green">E32</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_2_1"/>
( <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><a class="txt" href="prepower.html#E1:38"><i><font color="Green">E31</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1"><b>suppose </b></a><a NAME="E1:38_2_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_2_1_1"/>
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="prepower.html#E1:38"><i><font color="Green">E31</font></i></a>, <a class="txt" href="prepower.html#E2:38"><i><font color="Green">E32</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_2"><b>suppose </b></a><a NAME="E1:38_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_2_1_2"/>
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="prepower.html#E2:38"><i><font color="Green">E32</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:38"/><b>then </b>
<font color="Maroon">n</font> <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> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="prepower.html#E2:38"><i><font color="Green">E32</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E6:38"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="prepower.html#E1:38"><i><font color="Green">E31</font></i></a>, <a class="txt" href="prepower.html#E2:38"><i><font color="Green">E32</font></i></a>, </i>;<br/>

<a NAME="E7:38"/>
<font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="prepower.html#K5" target="_self">-Root</a> 1
 
<b>by </b><i><a class="txt" href="prepower.html#E1:38"><i><font color="Green">E31</font></i></a>, <a class="txt" href="prepower.html#E2:38"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E8:38"/>
<font color="Maroon">n</font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E2:38"><i><font color="Green">E32</font></i></a>, </i>;<br/>


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