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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">s</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>, <font color="Maroon">s1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green">E33</font></i>: 
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">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 0
 ;<br/>

<i><font color="Green">E34</font></i>: <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">s</font></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s1</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 = 
<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">s</font></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D12">CLVECT_1:def 12</a></i>
<br/>.= 
<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">s</font></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E1:40"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<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">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<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">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</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">s</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><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">s</font></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<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">s</font></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D12">CLVECT_1:def 12</a></i>
<br/>.= 
<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">s</font></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E1:40"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<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">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E1:40"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D12">CLVECT_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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">s</font> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:40"/>
 <a href="clopban3.html#K1" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="funct_2.html#R2">=</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">s</font></span>)</span> <a href="clopban3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="txt" href="clopban3.html#E2:40"><i><font color="Green">E34</font></i></a>, </i>;<br/>


</div>
