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<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">q</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:2"/><i><font color="Green">E26</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> )
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E2:2"/><i><font color="Green">E27</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E3:2"/><i><font color="Green">E28</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>case </b></a><a NAME="E1:2_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b><i><font color="Green">E30</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> = 
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">0 <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
0

;<br/><a NAME="E3:2_1_1_1"/>
<font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>case </b></a><a NAME="E1:2_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>


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