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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E31</font></i>: 
 <a href="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="seq_4.html#V1">bounded_above</a>
 ;<br/><b>reconsider </b><font color="Maroon">K</font> =  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E3:37_1"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_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="E1:37_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:37_1_1_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="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="csspace4.html#E1:37_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:37_1_1_1"/><i><font color="Green">E52</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>
 ;<br/><b>thus </b><a NAME="E5:37_1_1_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>
<b>consider </b><font color="Maroon">r0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#R2">in</a>  <a href="csspace4.html#K7" 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><a class="txt" href="csspace4.html#E1:37_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E5:37_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font>
 
<b>by </b><i>, <a class="txt" href="csspace4.html#E1:37_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:37_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="txt" href="csspace4.html#E1:37_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:37_1_2"/>
<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> <a href="hidden.html#R2">in</a>  <a href="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font>
 ;<br/><b>hence </b><a NAME="E2:37_1_2"/>
<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> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/></div><b>end;</b></div><b>take </b><font color="Maroon">K</font> = <font color="Maroon">K</font>;<br/><b>thus </b><a NAME="E5:37_1"/>
<font color="Maroon">g</font> is <a href="csspace4.html#V1" target="_self">bounded</a>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:37"/>
( <font color="Maroon">g</font> is <a href="csspace4.html#V1" target="_self">bounded</a> iff  <a href="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="seq_4.html#V1">bounded_above</a> )
 <b>by </b><i><a class="txt" href="csspace4.html#E1"><i><font color="Green">Lemma27</font></i></a></i>;<br/>


</div>
