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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/>







<a NAME="E1:45"/>
<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="ref" href="clvect_3.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E2:45"/><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#K3" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font>,<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>
 
;<br/>
<a NAME="E3:45"/><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#K3" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K3" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font>,<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>
 
;<br/>
<a NAME="E4:45"/><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#K3" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K3" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font>,<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"><a href="series_1.html#K4">Sum</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>,<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="ref" href="series_1.html#D6">SERIES_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:45"/>
<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#K3" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K3" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font>,<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"><a href="series_1.html#K4">Sum</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>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K4">Sum</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="series_1.html#D6">SERIES_1:def 6</a></i>;<br/>


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