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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rat_1.html#NM1">Rational</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:117"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a> )
 <b>and </b><br/><a NAME="E2:117"/><i><font color="Green">E96</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E3:117"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1"><b>suppose </b></a><a NAME="E1:117_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:117_1_1"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E1:117"><i><font color="Green">E95</font></i></a>, <a class="txt" href="prepower.html#E2:117"><i><font color="Green">E96</font></i></a>, <a class="txt" href="prepower.html#E3:117"><i><font color="Green">E97</font></i></a>, <a class="txt" href="prepower.html#E121"><i><font color="Green">Lemma93</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_2"><b>suppose </b></a><a NAME="E1:117_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:117_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K2">"</a> ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:117_1_2"/><i><font color="Green">E100</font></i>: 
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="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_2_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="E1:117_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E3:117"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E2:117_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T63" target="_self">Th63</a></i>;<br/><b>hence </b><a NAME="E3:117_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E3:117"><i><font color="Green">E97</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:117_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><a NAME="E7:117_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="prepower.html#E1:117"><i><font color="Green">E95</font></i></a>, <a class="ref" href="seq_2.html#T27">SEQ_2:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E9:117_1_2"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> ) )
 <b>by </b><i><a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E11:117_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="prepower.html#K9" target="_self">#Q</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#E9:117_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="prepower.html#T63" target="_self">Th63</a></i>;<br/><div><i><font color="Green">E104</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_1_2_2"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:117_1_2_2"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E1:117"><i><font color="Green">E95</font></i></a>, <a class="txt" href="prepower.html#E11:117_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_2_2_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/><b>assume </b><a NAME="E1:117_1_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:117_1_2_2_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E3:117_1_2_2"><i><font color="Green">E105</font></i></a></i>;<br/><a NAME="E3:117_1_2_2_1"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E3:117"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E4:117_1_2_2_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E3:117"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E5:117_1_2_2_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</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="prepower.html#T63" target="_self">Th63</a></i>;<br/><a NAME="E6:117_1_2_2_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E3:117_1_2_2_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E7:117_1_2_2_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E3:117"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E8:117_1_2_2_1"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E4:117_1_2_2_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="prepower.html#T63" target="_self">Th63</a></i>;<br/><a NAME="E9:117_1_2_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E9:117_1_2"><i><font color="Green">E102</font></i></a></i>;<br/><a NAME="E10:117_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <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>6</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E4:117_1_2_2_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E11:117_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E4:117_1_2_2_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E12:117_1_2_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E7:117_1_2_2_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E13:117_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E1:117_1_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="prepower.html#T72" target="_self">Th72</a></i>;<br/><a NAME="E14:117_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E9:117_1_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="prepower.html#E4:117_1_2_2_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="prepower.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E15:117_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E5:117_1_2_2_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E16:117_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E6:117_1_2_2_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="prepower.html#E8:117_1_2_2_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:117_1_2_2"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E13:117_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="prepower.html#E1:117"><i><font color="Green">E95</font></i></a>, <a class="txt" href="prepower.html#E6:117_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="seq_2.html#T35">SEQ_2:35</a></i>;<br/><a NAME="E14:117_1_2"/><i><font color="Green">E106</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
 <b>by </b><i><a class="txt" href="prepower.html#E1:117"><i><font color="Green">E95</font></i></a>, <a class="txt" href="prepower.html#E6:117_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="prepower.html#E12:117_1_2"><i><font color="Green">E104</font></i></a>, <a class="ref" href="seq_2.html#T36">SEQ_2:36</a></i>;<br/><a NAME="E15:117_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K18">"</a> 
 <b>by </b><i><a class="txt" href="prepower.html#E4:117_1_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:117_1_2_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E3:117"><i><font color="Green">E97</font></i></a></i>;<br/><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="prepower.html#E4:117_1_2"><i><font color="Green">E100</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="prepower.html#E3:117"><i><font color="Green">E97</font></i></a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:117_1_2_3"><i><font color="Green">E108</font></i></a>, <a class="ref" href="prepower.html#T65" target="_self">Th65</a></i>
;<br/><b>hence </b><a NAME="E3:117_1_2_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="prepower.html#E3:117"><i><font color="Green">E97</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E17:117_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E1:117"><i><font color="Green">E95</font></i></a>, <a class="txt" href="prepower.html#E2:117"><i><font color="Green">E96</font></i></a>, <a class="txt" href="prepower.html#E2:117_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="prepower.html#E13:117_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="prepower.html#E14:117_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="prepower.html#E15:117_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="prepower.html#E121"><i><font color="Green">Lemma93</font></i></a></i>;<br/><a NAME="E18:117_1_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/><a NAME="E19:117_1_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E2:117"><i><font color="Green">E96</font></i></a>, <a class="ref" href="prepower.html#T65" target="_self">Th65</a></i>;<br/><b>hence </b><a NAME="E20:117_1_2"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T59">XCMPLX_1:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
