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

<b>assume </b><a NAME="E1:21"/><i><font color="Green">E24</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="normsp_1.html#L1">NORMSTR</a> ;<br/>

<b>assume </b><a NAME="E2:21"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#G1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10">Zero_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="lp_space.html#K3" target="_self">l_norm^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 ;<br/>

<a NAME="E3:21"/><i><font color="Green">E26</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:21"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E4:21"/><i><font color="Green">E27</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E3:21"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E5:21"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E6:21"/><i><font color="Green">E29</font></i>: 
1 <a href="real_1.html#K6">/</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="lp_space.html#E4:21"><i><font color="Green">E27</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<a NAME="E7:21"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:21"><i><font color="Green">E24</font></i></a>, <a class="txt" href="lp_space.html#E2:21"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lp_space.html#T10" target="_self">Th10</a></i>;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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="E1:21_1"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</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> <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"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E2:21_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E3:21_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="lp_space.html#E4:21"><i><font color="Green">E27</font></i></a>, <a class="txt" href="lp_space.html#E1:21_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="lp_space.html#E3"><i><font color="Green">Lemma3</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:21"/><i><font color="Green">E32</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E10:21"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lp_space.html#E7:21"><i><font color="Green">E30</font></i></a>, <a class="txt" href="lp_space.html#E8:21"><i><font color="Green">E31</font></i></a>, <a class="ref" href="series_1.html#T21">SERIES_1:21</a></i>;<br/>
<a NAME="E11:21"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="lp_space.html#E4:21"><i><font color="Green">E27</font></i></a>, <a class="txt" href="lp_space.html#E9:21"><i><font color="Green">E32</font></i></a>, <a class="txt" href="lp_space.html#E3"><i><font color="Green">Lemma3</font></i></a></i>;<br/>
<b>thus </b><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"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</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>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></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>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lp_space.html#E1:21"><i><font color="Green">E24</font></i></a>, <a class="txt" href="lp_space.html#E2:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="lp_space.html#E22"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></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>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"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</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>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lp_space.html#E6:21"><i><font color="Green">E29</font></i></a>, <a class="txt" href="lp_space.html#E10:21"><i><font color="Green">E33</font></i></a>, <a class="txt" href="lp_space.html#E11:21"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lp_space.html#E4"><i><font color="Green">Lemma4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</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"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</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>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lp_space.html#E4:21"><i><font color="Green">E27</font></i></a>, <a class="txt" href="lp_space.html#E6:21"><i><font color="Green">E29</font></i></a>, <a class="txt" href="lp_space.html#E9:21"><i><font color="Green">E32</font></i></a>, <a class="ref" href="holder_1.html#T2">HOLDER_1:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="real_1.html#K4">*</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"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</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>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lp_space.html#E5:21"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</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"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</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>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/>


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