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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="E1:51"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>given </b><font color="Maroon">l</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><a NAME="E2:51"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">l1</font> = 2 as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1"><b>suppose </b></a><a NAME="E1:51_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:51_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</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/><a NAME="E3:51_1_1_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:51_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="prepower.html#K6">#Z</a> <font color="Maroon">l1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T48">PREPOWER:48</a></i>;<br/><b>thus </b><font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> 2</span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">l</font>
<b>by </b><i><a class="txt" href="power.html#E4:51_1_1_1"><i><font color="Green">E37</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">l</font>
<b>by </b><i><a class="txt" href="power.html#E2:51"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">l</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> 2</span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">l</font>
<b>by </b><i><a class="txt" href="power.html#E2:51"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="prepower.html#K6">#Z</a> <font color="Maroon">l1</font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">l</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="prepower.html#K6">#Z</a> <font color="Maroon">l1</font></span>)</span> <a href="prepower.html#K6">#Z</a> <font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="power.html#D1" target="_self">Def1</a>, <a class="txt" href="power.html#E1:51"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="prepower.html#K6">#Z</a> <span class="p1">(<span class="default"><font color="Maroon">l1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T55">PREPOWER:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="power.html#E4:51_1_1_1"><i><font color="Green">E37</font></i></a>, , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_2"><b>suppose </b></a><a NAME="E1:51_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:51_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</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="E3:51_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K6">#Z</a> <font color="Maroon">l1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="power.html#E1:51_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="prepower.html#T48">PREPOWER:48</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> 2</span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">l</font>
<b>by </b><i>, <a class="txt" href="power.html#E2:51_1_1_2"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">l</font>
<b>by </b><i><a class="txt" href="power.html#E2:51"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">l</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> 2</span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">l</font>
<b>by </b><i><a class="txt" href="power.html#E2:51"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6">#Z</a> <font color="Maroon">l1</font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">l</font>
<b>by </b><i><a class="txt" href="power.html#E1:51_1_1_2"><i><font color="Green">E38</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K6">#Z</a> <font color="Maroon">l1</font></span>)</span> <a href="prepower.html#K6">#Z</a> <font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="power.html#D1" target="_self">Def1</a>, <a class="txt" href="power.html#E1:51"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="prepower.html#K6">#Z</a> <span class="p1">(<span class="default"><font color="Maroon">l1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T55">PREPOWER:55</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="power.html#E1:51_1_1_2"><i><font color="Green">E38</font></i></a>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:51"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">k</font>
 ;<br/>


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