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<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>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14" title="CLOPBAN1:func.14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">f</font> as  <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> <b>by </b><i><a class="ref" href="clopban1.html#D8" target="_self" title="CLOPBAN1:def.8">Def8</a></i>;<br/>
<div><i><font color="Green" title="E43">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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>;<br/><b>assume </b><a NAME="E1:65_1"/><i><font color="Green" title="E44">A2</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="c4">g</font>
 ;<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="E3:65_1"/><i><font color="Green" title="E45">A3</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#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">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#K5" title="CLVECT_1:func.5">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c6">t</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</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#E1:65_1"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/><b>thus </b><a NAME="E4:65_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="clopban1.html#E3:65_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="clvect_1.html#T106" title="CLVECT_1:th.106">CLVECT_1:106</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:65"/><i><font color="Green" title="E44">A4</font></i>: 
( not  <a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c4">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="c4">g</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="clopban1.html#T30" target="_self" title="CLOPBAN1:th.30">Th30</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">r0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:65"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c5">r0</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="c4">g</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">r0</font> = <font color="Maroon" title="c5">r0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="clopban1.html#E5:65"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/>
<a NAME="E7:65"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r0</font>
 
<b>by </b><i><a class="txt" href="clopban1.html#E2:65"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="clopban1.html#E5:65"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/>
<a NAME="E8:65"/><b>then </b><i><font color="Green" title="E46">A6</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="clopban1.html#E3:65"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="clopban1.html#E5:65"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E9:65"/>
<span class="p1">(<span class="default"><a href="clopban1.html#K13" title="CLOPBAN1:func.13">BoundedLinearOperatorsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="clopban1.html#K12" title="CLOPBAN1:func.12">PreNorms</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="clopban1.html#T34" target="_self" title="CLOPBAN1:th.34">Th34</a></i>;<br/>
<b>hence </b><a NAME="E10:65"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="txt" href="clopban1.html#E8:65"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>;<br/>


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