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<b>let </b><font color="Maroon">X</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:37"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i/>
<br/>.= 
<font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D14">CLVECT_1:def 14</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="clvect_1.html#D14">CLVECT_1:def 14</a></i>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><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>assume </b><a NAME="E1:37_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>thus </b><a NAME="E2:37_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E1:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="clvect_1.html#D14">CLVECT_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D14">CLVECT_1:def 14</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D14">CLVECT_1:def 14</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:37"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:37"/>
 <a href="clopban3.html#K1" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <font color="Maroon">seq</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
