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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/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:32"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E1:32"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E4:32"/><b>then </b><i><font color="Green">E28</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E1:32"><i><font color="Green">E26</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self">Def3</a></i>;<br/>
<i><font color="Green">E30</font></i>: <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>2</sub></font> = 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T14" target="_self">Th14</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="prepower.html#E1:32"><i><font color="Green">E26</font></i></a>, <a class="txt" href="prepower.html#E33"><i><font color="Green">Lemma20</font></i></a></i>
;<br/>
<a NAME="E7:32"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:32"><i><font color="Green">E26</font></i></a>, <a class="txt" href="prepower.html#E4:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="prepower.html#E33"><i><font color="Green">Lemma20</font></i></a></i>;<br/>
<a NAME="E8:32"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E5:32"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E9:32"/><b>then </b><i><font color="Green">E32</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<a NAME="E10:32"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E1:32"><i><font color="Green">E26</font></i></a>, <a class="txt" href="prepower.html#E4:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self">Def3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="prepower.html#E2:32"><i><font color="Green">E27</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1"><b>suppose </b></a><a NAME="E1:32_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E1:32"><i><font color="Green">E26</font></i></a>, <a class="txt" href="prepower.html#E6:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="prepower.html#E9:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="prepower.html#E33"><i><font color="Green">Lemma20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_2"><b>suppose </b></a><a NAME="E1:32_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K4" target="_self">-Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E1:32"><i><font color="Green">E26</font></i></a>, <a class="txt" href="prepower.html#E6:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="prepower.html#E7:32"><i><font color="Green">E31</font></i></a>, <a class="txt" href="prepower.html#E10:32"><i><font color="Green">E33</font></i></a>, <a class="txt" href="prepower.html#E21"><i><font color="Green">Lemma11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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