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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">k</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">k</font> as   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="power.html#E1:47"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><b>suppose </b></a><a NAME="E1:47_1_1_1"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:47_1_1_1"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T49" title="PREPOWER:th.49">PREPOWER:49</a></i>;<br/><a NAME="E3:47_1_1_1"/>
(  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T40" title="RAT_1:th.40">RAT_1:40</a></i>;<br/><b>then </b><i><font color="Green" title="E35">A4</font></i>: <font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c3">p</font> = 
1 <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D5" title="PREPOWER:def.5">PREPOWER:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="power.html#E2:47_1_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="prepower.html#T30" title="PREPOWER:th.30">PREPOWER:30</a></i>
;<br/><b>thus </b><font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">k</font> = 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K12" title="PREPOWER:func.12">#R</a> <font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="power.html#E1:47_1_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="power.html#D2" target="_self" title="POWER:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="power.html#E1:47_1_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="power.html#E4:47_1_1_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="prepower.html#T88" title="PREPOWER:th.88">PREPOWER:88</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2"><b>suppose </b></a><a NAME="E1:47_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="ref" href="power.html#D2" target="_self" title="POWER:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:47"/>
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">k</font>
 ;<br/>


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