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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  holds <br/><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b1">k</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">k</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font> )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c3">l</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><a NAME="E2:48"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">l</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="int_1.html#D3" onmouseover="rs('int_1/D3')" onmouseout="rh()">INT_1:def 3</a>,<a class="ref" href="abian.html#D1" onmouseover="rs('abian/D1')" onmouseout="rh()">ABIAN:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_1_1"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> 2</span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="power.html#T33" target="_self" onmouseover="rs('power/T33')" onmouseout="rh()">Th33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" target="_self" onmouseover="rs('power/T46')" onmouseout="rh()">Th46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">l</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 2</span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" target="_self" onmouseover="rs('power/T46')" onmouseout="rh()">Th46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> 2</span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#D2" target="_self" onmouseover="rs('power/D2')" onmouseout="rh()">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> 2</span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> <font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#D2" target="_self" onmouseover="rs('power/D2')" onmouseout="rh()">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T45" onmouseover="rs('prepower/T45')" onmouseout="rh()">PREPOWER:45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="power.html#D2" target="_self" onmouseover="rs('power/D2')" onmouseout="rh()">Def2</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_3"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font></span><br/><div class="add"><a NAME="E2:48_1_3"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T58" onmouseover="rs('xreal_1/T58')" onmouseout="rh()">XREAL_1:58</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 2</span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="power.html#T33" target="_self" onmouseover="rs('power/T33')" onmouseout="rh()">Th33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" target="_self" onmouseover="rs('power/T46')" onmouseout="rh()">Th46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">l</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> 2</span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" target="_self" onmouseover="rs('power/T46')" onmouseout="rh()">Th46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> 2</span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#D2" target="_self" onmouseover="rs('power/D2')" onmouseout="rh()">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> 2</span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> <font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#D2" target="_self" onmouseover="rs('power/D2')" onmouseout="rh()">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T45" onmouseover="rs('prepower/T45')" onmouseout="rh()">PREPOWER:45</a></span>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="power.html#D2" target="_self" onmouseover="rs('power/D2')" onmouseout="rh()">Def2</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
