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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:10"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="comptrig.html#E2:10"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E4:10"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="rat_1.html#NM1">Rational</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="comptrig.html#E1:10"><i><font color="Green">E17</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>suppose </b></a><a NAME="E1:10_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:10_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="power.html#K1">-root</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4">-Root</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="comptrig.html#E1:10"><i><font color="Green">E17</font></i></a>, <a class="txt" href="comptrig.html#E4:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="power.html#D1">POWER:def 1</a></i>;<br/><a NAME="E3:10_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="power.html#K1">-root</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="comptrig.html#E4:10"><i><font color="Green">E20</font></i></a>, <a class="txt" href="comptrig.html#E1:10_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="prepower.html#D3">PREPOWER:def 3</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K1">-root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K1">-root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K1">-root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prepower.html#K8">#Q</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="comptrig.html#E3:10_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="prepower.html#T88">PREPOWER:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K1">-root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="comptrig.html#E3:10_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="prepower.html#T60">PREPOWER:60</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="comptrig.html#E1:10"><i><font color="Green">E17</font></i></a>, <a class="txt" href="comptrig.html#E4:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="power.html#T5">POWER:5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2"><b>suppose </b></a><a NAME="E1:10_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="power.html#K1">-root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font> = 
0 <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="comptrig.html#E4:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="power.html#T6">POWER:6</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="comptrig.html#E3:10"><i><font color="Green">E19</font></i></a>, <a class="txt" href="comptrig.html#E1:10_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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