<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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>3</sub></font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>





<b>assume </b><a NAME="E1:118"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="prepower.html#NM1">Rational_Sequence</a><b> such that </b><br/><a NAME="E3:118"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="prepower.html#T80" target="_self">Th80</a></i>;<br/>
<a NAME="E4:118"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:118"><i><font color="Green">E96</font></i></a>, <a class="txt" href="prepower.html#E3:118"><i><font color="Green">E97</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self">Th82</a></i>;<br/>
<a NAME="E5:118"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K12" target="_self">#R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:118"><i><font color="Green">E96</font></i></a>, <a class="txt" href="prepower.html#E3:118"><i><font color="Green">E97</font></i></a>, <a class="ref" href="prepower.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E6:118"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>4</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#E1:118"><i><font color="Green">E96</font></i></a>, <a class="ref" href="prepower.html#T95" target="_self">Th95</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:118_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/><b>hence </b><a NAME="E2:118_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="rat_1.html#NM1">Rational</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="prepower.html#NM1">Rational_Sequence</a> <b>by </b><i><a class="ref" href="prepower.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E9:118"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E3:118"><i><font color="Green">E97</font></i></a>, <a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/>
<a NAME="E10:118"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>5</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:118"><i><font color="Green">E96</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self">Th82</a></i>;<br/>
<a NAME="E11:118"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E3:118"><i><font color="Green">E97</font></i></a>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>;<br/>
<a NAME="E12:118"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K12" target="_self">#R</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:118"><i><font color="Green">E96</font></i></a>, <a class="txt" href="prepower.html#E9:118"><i><font color="Green">E101</font></i></a>, <a class="txt" href="prepower.html#E10:118"><i><font color="Green">E102</font></i></a>, <a class="ref" href="prepower.html#D8" target="_self">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><i><font color="Green">E104</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="prepower.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:118"><i><font color="Green">E96</font></i></a>, <a class="ref" href="prepower.html#T70" target="_self">Th70</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="prepower.html#D7" target="_self">Def7</a></i>
;<br/><a NAME="E2:118_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>6</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#E1:118"><i><font color="Green">E96</font></i></a>, <a class="ref" href="prepower.html#T63" target="_self">Th63</a></i>;<br/><b>hence </b><a NAME="E3:118_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="prepower.html#E1:118_2"><i><font color="Green">E104</font></i></a>, <a class="ref" href="prepower.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:118"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K12" target="_self">#R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K12" target="_self">#R</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E4:118"><i><font color="Green">E98</font></i></a>, <a class="txt" href="prepower.html#E5:118"><i><font color="Green">E99</font></i></a>, <a class="txt" href="prepower.html#E6:118"><i><font color="Green">E100</font></i></a>, <a class="txt" href="prepower.html#E10:118"><i><font color="Green">E102</font></i></a>, <a class="txt" href="prepower.html#E12:118"><i><font color="Green">E103</font></i></a>, <a class="ref" href="prepower.html#T103" target="_self">Th103</a></i>;<br/>


</div>
