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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being <a href="clopban1.html#V3" title="CLOPBAN1:attr.3">bounded</a> <a href="clopban1.html#NM1" title="CLOPBAN1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> holds <br/> ( not  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Olive" title="b1">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Olive" title="b1">g</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> )</span><br/><b>let </b><font color="Maroon" title="c3">g</font> be  <a href="clopban1.html#V3" title="CLOPBAN1:attr.3">bounded</a> <a href="clopban1.html#NM1" title="CLOPBAN1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c3">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c3">g</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> )</span><br/>



<a NAME="E1:53"/>
 <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c3">g</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c4">K</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:53_1"/><i><font color="Green" title="E33">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">K</font>
 <b>and </b><br/><a NAME="E3:53_1"/><i><font color="Green" title="E34">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">K</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <b>by </b><i><a class="ref" href="clopban1.html#D7" target="_self" title="CLOPBAN1:def.7">Def7</a></i>;<br/>
<div><i><font color="Green" title="E35">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c3">g</font> implies <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">K</font> )</span><br/><b>assume </b><a NAME="E1:53_1_1"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">K</font></span><br/><b>consider </b><font color="Maroon" title="c6">t</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:53_1_1"/><i><font color="Green" title="E37">A5</font></i>: 
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c6">t</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="clopban1.html#E1:53_1_1"><i><font color="Green" title="E36">A4</font></i></a></i>;<br/><a NAME="E4:53_1_1"/><i><font color="Green" title="E38">A6</font></i>: 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">K</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c6">t</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <b>by </b><i><a class="txt" href="clopban1.html#E3:53_1"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/><a NAME="E5:53_1_1"/>
<font color="Maroon" title="c4">K</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c6">t</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">K</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <b>by </b><i><a class="txt" href="clopban1.html#E2:53_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="clopban1.html#E3:53_1_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E6:53_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="txt" href="clopban1.html#E3:53_1_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="clopban1.html#E4:53_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c4">K</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c3">g</font> or <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">K</font> )</span><br/>

<b>thus </b><a NAME="E5:53_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c3">g</font> or <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">K</font> )
 <b>by </b><i><a class="txt" href="clopban1.html#E4:53_1"><i><font color="Green" title="E35">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:53"/>
( not  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c3">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c3">g</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
