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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>;<br/>
<b>consider </b><font color="Maroon">IT</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:2_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:2_1_1"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:2_1_1_1"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <a href="rsspace.html#K6">Zeroseq</a> </span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:2_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">rseq</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <a href="rsspace.html#K6">Zeroseq</a> </span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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"><a href="rsspace.html#K2">seq_id</a> <a href="rsspace.html#K6">Zeroseq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> 0</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="rsspace.html#D6">RSSPACE:def 6</a></i>
<br/>.= 
0 <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:2_1_1_1_1"/><b>then </b>
<font color="Maroon">rseq</font> is <a href="series_1.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T3">COMSEQ_3:3</a></i>;<br/>
<b>hence </b><a NAME="E5:2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <a href="rsspace.html#K6">Zeroseq</a> </span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#T40">SERIES_1:40</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:2_1_1_1"/>
not <font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:2_1_1"/>
<font color="Maroon">IT</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:2_1_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:2_1_1_2"/>
<font color="Maroon">IT</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:2_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a>  &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">x</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a> ) )
 <b>by </b><i>, , <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>;<br/>


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