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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> = 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="csspace.html#T44">CSSPACE:44</a></i>
;<br/><a NAME="E3:44_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E4:44_1"/><i><font color="Green">E25</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> is <a href="seqm_3.html#V3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="clvect_3.html#T35" target="_self">Th35</a></i>;<br/><div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:44_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>]
 ;<br/><span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> = 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_3.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/><a NAME="E3:44_1_2"/><b>then </b><i><font color="Green">E28</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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></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"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></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="E4:44_1_2"/>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></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"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <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"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clvect_3.html#E1:44_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:44_1_2"/><b>then </b><i><font color="Green">E29</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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></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"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <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"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clvect_3.html#E3:44_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:44_1_2"/>
<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">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="clvect_3.html#E4:44_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="seqm_3.html#T11">SEQM_3:11</a></i>;<br/><a NAME="E7:44_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<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">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E8:44_1_2"/><b>then </b><i><font color="Green">E31</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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></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"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clvect_3.html#E5:44_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E9:44_1_2"/>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="csspace.html#T46">CSSPACE:46</a></i>;<br/><a NAME="E10:44_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="clvect_3.html#E7:44_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18">abs</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="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="csspace.html#K13">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_2.html#D3">CLVECT_2:def 3</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E12:44_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="clvect_3.html#E8:44_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="clvect_3.html#E10:44_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E6:44_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="clvect_3.html#E3:44_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="clvect_3.html#E5:44_1"><i><font color="Green">E26</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="clvect_3.html#E1:44_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E9:44_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>hence </b><a NAME="E10:44_1"/>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clvect_3.html#E6:44_1"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span>;<br/><span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> = 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="csspace.html#T44">CSSPACE:44</a></i>
;<br/><a NAME="E3:44_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E4:44_2"/><i><font color="Green">E25</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> is <a href="seqm_3.html#V3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="clvect_3.html#T35" target="_self">Th35</a></i>;<br/><div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:44_2_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>]
 ;<br/><span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> = 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_3.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>

;<br/><a NAME="E3:44_2_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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></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"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="E4:44_2_2"/><b>then </b><i><font color="Green">E28</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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></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">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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#T49">CSSPACE:49</a></i>;<br/><a NAME="E5:44_2_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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></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">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></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">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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="txt" href="clvect_3.html#E1:44_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:44_2_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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p0"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p0"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></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">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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="txt" href="clvect_3.html#E4:44_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:44_2_2"/><b>then </b><i><font color="Green">E29</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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></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">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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="complex1.html#T138">COMPLEX1:138</a></i>;<br/><a NAME="E8:44_2_2"/>
<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">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="clvect_3.html#E4:44_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="seqm_3.html#T11">SEQM_3:11</a></i>;<br/><a NAME="E9:44_2_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<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">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E10:44_2_2"/><b>then </b><i><font color="Green">E31</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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></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">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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="txt" href="clvect_3.html#E7:44_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E11:44_2_2"/>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="csspace.html#T46">CSSPACE:46</a></i>;<br/><a NAME="E12:44_2_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></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">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="clvect_3.html#E9:44_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</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="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="csspace.html#K13">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_2.html#D3">CLVECT_2:def 3</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</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="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span>

;<br/><b>hence </b><a NAME="E14:44_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="clvect_3.html#E10:44_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="clvect_3.html#E12:44_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E6:44_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="clvect_3.html#E3:44_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="clvect_3.html#E5:44_2"><i><font color="Green">E26</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="clvect_3.html#E1:44_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E9:44_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>hence </b><a NAME="E10:44_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"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clvect_3.html#E6:44_2"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:44"/>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clvect_3.html#E1:44"><i><font color="Green">E22</font></i></a></i>;<br/>


</div>
