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<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>let </b><font color="Maroon">k</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a> <a href="int_1.html#NM1">Integer</a>;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_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="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><a NAME="E1:8_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:8_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="power.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T38">POWER:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><a NAME="E1:8_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:8_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="xcmplx_0.html#K4">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:8_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E4:8_1_2"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="power.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:8_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">k</font> <a href="power.html#K3">to_power</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><font color="Maroon">k</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">k</font> <a href="prepower.html#K6">#Z</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<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">k</font> <a href="prepower.html#K6">#Z</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K6">#Z</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="prepower.html#T55">PREPOWER:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="power.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K6">#Z</a> <font color="Maroon">n</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">k</font> <a href="power.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/><b>hence </b><a NAME="E7:8_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="power.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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