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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">k</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><a NAME="E1:53"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:53_1_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> 
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font></span>)</span> = 
1 <a href="real_1.html#K6">/</a> 1
<b>by </b><i/>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K2" target="_self">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3"><b>suppose </b></a><a NAME="E1:53_1_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:53_1_3"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> 
<b>by </b><i>, </i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E4:53_1_3"/>
<font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6" target="_self">#Z</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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