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<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:4"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 ) )
 ;<br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:4"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i/>;<br/>
<a NAME="E5:4"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="real_2.html#T137">REAL_2:137</a></i>;<br/>
<a NAME="E6:4"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1"><b>case </b></a><a NAME="E1:4_1_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="power.html#K2">-root</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K2">-root</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="power.html#T5">POWER:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2"><b>case </b></a><a NAME="E1:4_1_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:4_1_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:4_1_1_2"/>
<font color="Maroon">n</font> <a href="power.html#K2">-root</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="power.html#K1">-root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="power.html#T1">POWER:1</a></i>;<br/><a NAME="E4:4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K2">-root</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="power.html#T5">POWER:5</a></i>;<br/><b>hence </b><a NAME="E5:4_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="power.html#K2">-root</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K2">-root</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="power.html#K2">-root</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K2">-root</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/>


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