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<b>let </b><font color="Maroon">X</font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</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>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"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_2.html#D3">CLVECT_2:def 3</a></i>
;<br/>
<a NAME="E2:42"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_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="E1:42_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:42_2"/>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.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="p1">(<span class="default"><font color="Maroon">seq</font> <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/><a NAME="E3:42_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <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><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.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><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><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><a href="csspace.html#K13">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace.html#T48">CSSPACE:48</a></i>;<br/><a NAME="E4:42_2"/>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.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><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><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><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><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><a href="csspace.html#K13">.||</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:42_2"/><b>then </b>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <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><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><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><a href="csspace.html#K13">.||</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:42_2"/><b>then </b>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <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><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">seq</font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_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="ref" href="clvect_2.html#D3">CLVECT_2:def 3</a></i>;<br/><b>hence </b><a NAME="E7:42_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E4:42"/>
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>(, );</i><br/>


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