<?xml version="1.0"?>
<div class="add">

<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">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:87"/>
<font color="Maroon" title="c3">seq</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">M</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:87"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c4">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E4:87"/><i><font color="Green" title="E46">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds  <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">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> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="clvect_2.html#D10" target="_self" title="CLVECT_2:def.10">Def10</a></i>;<br/>
<a NAME="E5:87"/><i><font color="Green" title="E47">A3</font></i>: 
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies <font color="Maroon" title="c2">z</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">*</a> <font color="Maroon" title="c3">seq</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:87_1"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</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/><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"><font color="Maroon" title="c2">z</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> = 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<b>by </b><i><a class="txt" href="clvect_2.html#E1:87_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="clvect_1.html#D14" title="CLVECT_1:def.14">CLVECT_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">X</font>)</span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#T2" title="CLVECT_1:th.2">CLVECT_1:2</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="csspace.html#T44" title="CSSPACE:th.44">CSSPACE:44</a></i>
;<br/><b>hence </b><a NAME="E2:87_1_1"/>
<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"><font color="Maroon" title="c2">z</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">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> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E3:87"><i><font color="Green" title="E45">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:87_1"/>
<font color="Maroon" title="c2">z</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">*</a> <font color="Maroon" title="c3">seq</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="clvect_2.html#E3:87"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="clvect_2.html#D10" target="_self" title="CLVECT_2:def.10">Def10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:87"/>
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 implies <font color="Maroon" title="c2">z</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">*</a> <font color="Maroon" title="c3">seq</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:87_2"/>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:87_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/>
<a NAME="E3:87_2"/><b>then </b><i><font color="Green" title="E48">A5</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="clvect_2.html#E3:87"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</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="E1:87_2_1"/><i><font color="Green" title="E49">A6</font></i>: 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">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> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E4:87"><i><font color="Green" title="E46">A2</font></i></a></i>;<br/><i><font color="Green" title="E50">A7</font></i>: <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"><font color="Maroon" title="c2">z</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> = 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D14" title="CLVECT_1:def.14">CLVECT_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<b>by </b><i><a class="ref" href="csspace.html#T45" title="CSSPACE:th.45">CSSPACE:45</a></i>
;<br/><a NAME="E3:87_2_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E4:87_2_1"/>
<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"><font color="Maroon" title="c2">z</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">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> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E1:87_2_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="clvect_2.html#E2:87_2_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:87_2"/>
<font color="Maroon" title="c2">z</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">*</a> <font color="Maroon" title="c3">seq</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="clvect_2.html#E3:87_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="clvect_2.html#D10" target="_self" title="CLVECT_2:def.10">Def10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:87"/>
<font color="Maroon" title="c2">z</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">*</a> <font color="Maroon" title="c3">seq</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="clvect_2.html#E5:87"><i><font color="Green" title="E47">A3</font></i></a></i>;<br/>


</div>
