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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> $1 is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>;<br/>
<b>consider </b><font color="Maroon" title="c1">IT</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:1_1_1"/><i><font color="Green" title="E1">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:1_1_1"/>
<font color="Maroon" title="c1">IT</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> 
 
<b>by </b><i><a class="txt" href="csspace3.html#E2:1_1_1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:1_1_1_1"/>
<font color="Maroon" title="c1">IT</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a>  st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> iff ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  &amp;  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">x</font> is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a> ) )
 <b>by </b><i><a class="txt" href="csspace3.html#E2:1_1_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="csspace.html#D7" title="CSSPACE:def.7">CSSPACE:def 7</a></i>;<br/>


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