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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:110"/><i><font color="Green">E89</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> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:110"/><b>then </b><i><font color="Green">E90</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="rat_1.html#NM1">Rational</a><b> such that </b><br/><a NAME="E4:110"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="prepower.html#E1:110"><i><font color="Green">E89</font></i></a>, <a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="rat_1.html#NM1">Rational</a><b> such that </b><br/><a NAME="E6:110"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="prepower.html#E4:110"><i><font color="Green">E91</font></i></a>, <a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/>
<a NAME="E7:110"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:110"><i><font color="Green">E89</font></i></a>, <a class="txt" href="prepower.html#E6:110"><i><font color="Green">E92</font></i></a>, <a class="ref" href="prepower.html#T75" target="_self">Th75</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="prepower.html#NM1">Rational_Sequence</a><b> such that </b><br/><a NAME="E9:110"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>6</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>6</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>6</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/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="prepower.html#NM1">Rational_Sequence</a><b> such that </b><br/><a NAME="E11:110"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>7</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>7</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="ref" href="prepower.html#T79" target="_self">Th79</a></i>;<br/>
<a NAME="E12:110"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>6</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E2:110"><i><font color="Green">E90</font></i></a>, <a class="txt" href="prepower.html#E9:110"><i><font color="Green">E94</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self">Th82</a></i>;<br/>
<a NAME="E13:110"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">c<sub>7</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E2:110"><i><font color="Green">E90</font></i></a>, <a class="txt" href="prepower.html#E11:110"><i><font color="Green">E95</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self">Th82</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:110_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>8</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="txt" href="prepower.html#E9:110"><i><font color="Green">E94</font></i></a></i>;<br/><a NAME="E2:110_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E4:110"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:110_1"/><b>then </b>
<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>6</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E1:110"><i><font color="Green">E89</font></i></a>, <a class="ref" href="prepower.html#T74" target="_self">Th74</a></i>;<br/><b>hence </b><a NAME="E4:110_1"/>
<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>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="prepower.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:110"/><b>then </b>
 <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>6</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E12:110"><i><font color="Green">E96</font></i></a>, <a class="ref" href="prepower.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E16:110"/><b>then </b><i><font color="Green">E98</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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E2:110"><i><font color="Green">E90</font></i></a>, <a class="txt" href="prepower.html#E9:110"><i><font color="Green">E94</font></i></a>, <a class="txt" href="prepower.html#E12:110"><i><font color="Green">E96</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="110_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:110_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E11:110"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E2:110_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E6:110"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:110_2"/><b>then </b>
<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>7</sub></font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E1:110"><i><font color="Green">E89</font></i></a>, <a class="ref" href="prepower.html#T74" target="_self">Th74</a></i>;<br/><b>hence </b><a NAME="E4:110_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>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="prepower.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:110"/><b>then </b>
 <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>7</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E13:110"><i><font color="Green">E97</font></i></a>, <a class="ref" href="prepower.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E19:110"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K12" target="_self">#R</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E2:110"><i><font color="Green">E90</font></i></a>, <a class="txt" href="prepower.html#E11:110"><i><font color="Green">E95</font></i></a>, <a class="txt" href="prepower.html#E13:110"><i><font color="Green">E97</font></i></a>, <a class="ref" href="prepower.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E20:110"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K12" target="_self">#R</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E7:110"><i><font color="Green">E93</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E21:110"/>
<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="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K12" target="_self">#R</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E16:110"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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