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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">f</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14" target="_self">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as  <a href="clopban1.html#V3" target="_self">bounded</a> <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<div><i><font color="Green">E38</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">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E44</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/><a NAME="E2:65_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <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="Olive">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> where <font color="Olive">t</font> is   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> : <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Olive">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="clopban1.html#D1" target="_self">Def1</a></i>;<br/><b>then 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="E4:65_1"/><i><font color="Green">E70</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 )
 ;<br/><b>thus </b><a NAME="E5:65_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>
<a NAME="E3:65"/><i><font color="Green">E84</font></i>: 
( not  <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <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/>
<b>consider </b><font color="Maroon">r0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:65"/><i><font color="Green">E85</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/>;<br/>
<a NAME="E7:65"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:65"/><b>then </b><i><font color="Green">E86</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <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>
 
<b>by </b><i><a class="ref" href="clopban1.html#D1" target="_self">Def1</a>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E9:65"/>
<span class="p1">(<span class="default"><a href="clopban1.html#K13" target="_self">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <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>
 
<b>by </b><i><a class="ref" href="clopban1.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E10:65"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">f</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, <a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>;<br/>


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