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<b>let </b><font color="Maroon">X</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>, <font color="Maroon">Y</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E38</font></i>: 
 <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="seq_4.html#V1">bounded_above</a>
 ;<br/><b>reconsider </b><font color="Maroon">K</font> =  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E3:57_1"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:57_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font>
 ;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:57_1_1_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="clopban1.html#T1" target="_self">Th1</a></i>;<br/><b>thus </b><a NAME="E4:57_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="clvect_1.html#T106">CLVECT_1:106</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">r0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:57_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r0</font> = <font color="Maroon">r0</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="clopban1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:57_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font>
 
<b>by </b><i>, <a class="ref" href="clopban1.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E6:57_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="clopban1.html#T1" target="_self">Th1</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> or <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1_1"><b>case </b></a><a NAME="E1:57_1_2_1_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E91</font></i>: <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="complex1.html#K5">0c</a>  <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#T2">CLVECT_1:2</a></i>
<br/>.= 
<a href="complex1.html#K5">0c</a>  <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="clvect_1.html#T2">CLVECT_1:2</a></i>
;<br/><a NAME="E3:57_1_2_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="clopban1.html#T1" target="_self">Th1</a>, <a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/><b>hence </b><a NAME="E4:57_1_2_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, <a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1_2"><b>case </b></a><a NAME="E1:57_1_2_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:57_1_2_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/><a NAME="E3:57_1_2_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="clvect_1.html#T106">CLVECT_1:106</a></i>;<br/><i><font color="Green">E94</font></i>: <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T15">ABSVALUE:15</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K2">"</a> 

;<br/><b>reconsider </b><font color="Maroon">t0</font> = <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">t0</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">t</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> ;<br/><i><font color="Green">E95</font></i>: <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t1</font></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">t0</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="clopban1.html#T1" target="_self">Th1</a>, , <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
;<br/><i><font color="Green">E96</font></i>: <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">t0</font></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i/>
;<br/><a NAME="E9:57_1_2_1_1_2"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="txt" href="clopban1.html#E3:57_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E10:57_1_2_1_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>

<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="clopban1.html#T1" target="_self">Th1</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>

;<br/><b>hence </b><a NAME="E12:57_1_2_1_1_2"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, <a class="ref" href="clopban1.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:57_1_2"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span>
 ;<br/></div><b>end;</b></div><b>take </b><font color="Maroon">K</font> = <font color="Maroon">K</font>;<br/><b>thus </b><a NAME="E5:57_1"/>
<font color="Maroon">g</font> is <a href="clopban1.html#V3" target="_self">bounded</a>
 <b>by </b><i><a class="txt" href="clopban1.html#E1:57_1"><i><font color="Green">E38</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:57"/>
( <font color="Maroon">g</font> is <a href="clopban1.html#V3" target="_self">bounded</a> iff  <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="seq_4.html#V1">bounded_above</a> )
 <b>by </b><i/>;<br/>


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