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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green">E10</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font> is <a href="series_1.html#V1">summable</a>
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font> is <a href="series_1.html#V1">summable</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><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="E1:9_1"/><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E2:9_1"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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"><a href="complex1.html#K18">abs</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>7</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a></i>;<br/><i><font color="Green">E16</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <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>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
;<br/><b>thus </b><a NAME="E4:9_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lp_space.html#E1:9_1"><i><font color="Green">E14</font></i></a></i>;<br/><b>thus </b><a NAME="E5:9_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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"><a href="complex1.html#K18">abs</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>7</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lp_space.html#E2:9_1"><i><font color="Green">E15</font></i></a></i>;<br/><b>thus </b><a NAME="E6:9_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <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>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lp_space.html#E3:9_1"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="lp_space.html#E1:9"><i><font color="Green">E10</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="9_2_1_1"><b>case </b></a><a NAME="E1:9_2_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:9_2_1_1"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font> is <a href="series_1.html#V1">summable</a> )
 <b>by </b><i><a class="txt" href="lp_space.html#E2:9"><i><font color="Green">E11</font></i></a>, <a class="txt" href="lp_space.html#E3:9"><i><font color="Green">E12</font></i></a>, <a class="txt" href="lp_space.html#E4:9"><i><font color="Green">E13</font></i></a>, <a class="txt" href="lp_space.html#E1:9_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="holder_1.html#T13">HOLDER_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2"><b>case </b></a><a NAME="E1:9_2_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:9_2_1_2"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font> is <a href="series_1.html#V1">summable</a> )
 <b>by </b><i><a class="txt" href="lp_space.html#E2:9"><i><font color="Green">E11</font></i></a>, <a class="txt" href="lp_space.html#E3:9"><i><font color="Green">E12</font></i></a>, <a class="txt" href="lp_space.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="lp_space.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:9"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font> is <a href="series_1.html#V1">summable</a> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 ;<br/>


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