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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="rat_1.html#NM1">Rational</a>;<br/>



<b>assume </b><a NAME="E1:78"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:78"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="rat_1.html#K2">numerator</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rat_1.html#T77">RAT_1:77</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="rat_1.html#K2">numerator</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E4:78"/><i><font color="Green">E66</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="rat_1.html#T29">RAT_1:29</a></i>;<br/>
<a NAME="E5:78"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="prepower.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E6:78"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="prepower.html#E2:78"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:78"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:78"><i><font color="Green">E64</font></i></a>, <a class="txt" href="prepower.html#E21"><i><font color="Green">Lemma11</font></i></a></i>;<br/>
<a NAME="E8:78"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="prepower.html#E5:78"><i><font color="Green">E67</font></i></a>, <a class="ref" href="newton.html#T15">NEWTON:15</a></i>;<br/>
<a NAME="E9:78"/><b>then </b>
<span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prepower.html#K5" target="_self">-Root</a> 1
 
<b>by </b><i><a class="txt" href="prepower.html#E4:78"><i><font color="Green">E66</font></i></a>, <a class="ref" href="prepower.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E10:78"/><b>then </b>
<span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prepower.html#K4" target="_self">-Root</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K6" target="_self">#Z</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="prepower.html#E4:78"><i><font color="Green">E66</font></i></a>, <a class="ref" href="prepower.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E11:78"/>
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>


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