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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>







<a NAME="E1:46"/>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K3" title="CLVECT_3:func.3">Sum</a> <font color="Maroon" title="c2">seq</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K3" title="CLVECT_3:func.3">Sum</a> <font color="Maroon" title="c2">seq</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p3"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p3"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="clvect_3.html#T40" target="_self" title="CLVECT_3:th.40">Th40</a></i>;<br/>
<a NAME="E2:46"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K4" title="CLVECT_3:func.4">Sum</a> <font color="Maroon" title="c2">seq</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c3">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p3"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p3"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E3:46"/>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K4" title="CLVECT_3:func.4">Sum</a> <font color="Maroon" title="c2">seq</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c3">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K5" title="SERIES_1:func.5">Sum</a> <span class="p2"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="series_1.html#D7" title="SERIES_1:def.7">SERIES_1:def 7</a></i>;<br/>


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