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<b>let </b><font color="Maroon">X</font> be  non <a href="realset2.html#V3">trivial</a> <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/>

<b>consider </b><font color="Maroon">v</font> being   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:39"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">v</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ii</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> as  <a href="clopban1.html#V3">bounded</a> <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E4:39"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">ii</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_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="E1:39_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K12">PreNorms</a> <font color="Maroon">ii</font>
 ;<br/><a NAME="E2:39_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">ii</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/>;<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="E4:39_1"/><i><font color="Green">E53</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">ii</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="E5:39_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:39"/><i><font color="Green">E55</font></i>: 
( ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K12">PreNorms</a> <font color="Maroon">ii</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) implies  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="clopban1.html#K12">PreNorms</a> <font color="Maroon">ii</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<a NAME="E7:39"/>
1 <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">ii</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>, </i>;<br/>
<a NAME="E8:39"/><b>then </b><i><font color="Green">E56</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K12">PreNorms</a> <font color="Maroon">ii</font>
 
;<br/>
<a NAME="E9:39"/>
( not  <a href="clopban1.html#K12">PreNorms</a> <font color="Maroon">ii</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="clopban1.html#K12">PreNorms</a> <font color="Maroon">ii</font> is <a href="seq_4.html#V1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="clopban1.html#T30">CLOPBAN1:30</a></i>;<br/>
<a NAME="E10:39"/><b>then </b>
1 <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">PreNorms</a> <font color="Maroon">ii</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E11:39"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="clopban1.html#K12">PreNorms</a> <font color="Maroon">ii</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E12:39"/>
<span class="p1">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="clopban1.html#T34">CLOPBAN1:34</a></i>;<br/>


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