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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/>

<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">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 or <font color="Maroon">c<sub>1</sub></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="51_1_1_1"><b>suppose </b></a><a NAME="E1:51_1_1_1"/><i><font color="Green">E34</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"><b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> 2 = 
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:51_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="power.html#T32" target="_self">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="power.html#T30" target="_self">Th30</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="power.html#T30" target="_self">Th30</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> 

;<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"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="power.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_3"><b>suppose </b></a><a NAME="E1:51_1_1_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = 1 as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> 2 = 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K6">#Z</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:51_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="power.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K6">#Z</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K6">#Z</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:51_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="prepower.html#T54">PREPOWER:54</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K6">#Z</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> 

;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:51"/>
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> 
 ;<br/>


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