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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="clopban1.html#V2" title="CLOPBAN1:attr.2">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="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="clopban1.html#V2" title="CLOPBAN1:attr.2">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )</span><br/>



<a NAME="E1:55"/>
 <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c3">g</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">K</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:55_1"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E3:55_1"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#D7" target="_self" title="CLOPBAN1:def.7">Def7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">K</font>
; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">assume </span><a NAME="E1:55_1_1"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:55_1_1"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:55_1_1"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<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
 ;<br/><a NAME="E5:55_1_1"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a></span>;<br/><a NAME="E6:55_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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E4:55_1_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><span class="kw">hence </span><a NAME="E7:55_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E5:55_1_1"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:55_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:55"/>
( 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="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
