<?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">f</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">g</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>







<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E33</font></i>: 
<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> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as  <a href="csspace4.html#V1" target="_self">bounded</a> <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> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">z</font> = <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span> as   <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> <b>by </b><i><a class="ref" href="funcop_1.html#T58">FUNCOP_1:58</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><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="E1:49_1_1"/>
<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> <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="txt" href="csspace4.html#E1:49_1"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E2:49_1_1"/><b>then </b>
<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#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="clvect_1.html#T106">CLVECT_1:106</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E5:49_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E6:49_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:49_2"/>
<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> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">z</font> = <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span> as   <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> <b>by </b><i><a class="ref" href="funcop_1.html#T58">FUNCOP_1:58</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as  <a href="csspace4.html#V1" target="_self">bounded</a> <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> <b>by </b><i/>;<br/>
<a NAME="E3:49_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">z</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_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="E1:49_2_1_1"/><i><font color="Green">E54</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="E2:49_2_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/>;<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="E4:49_2_1_1"/><i><font color="Green">E78</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/><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> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>
;<br/><b>hence </b><a NAME="E6:49_2_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E5:49_2_1"/><i><font color="Green">E79</font></i>: 
( not  <a href="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <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/>
<b>consider </b><font color="Maroon">r0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:49_2_1"/><i><font color="Green">E80</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/>;<br/>
<a NAME="E9:49_2_1"/><i><font color="Green">E82</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E4"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/>
<a NAME="E10:49_2_1"/>
( ( 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="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 ) implies  <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> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<a NAME="E11:49_2_1"/><b>then </b>
 <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> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="csspace4.html#E4"><i><font color="Green">Lemma42</font></i></a>, , , , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E12:49_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="csspace4.html#K8" target="_self">ComplexBoundedFunctionsNorm</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> 0
 
<b>by </b><i><a class="txt" href="csspace4.html#E1:49"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:49_2_1"/>
<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> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:49"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font>, <font color="Maroon">h1</font> = <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font> as  <a href="csspace4.html#V1" target="_self">bounded</a> <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> <b>by </b><i/>;<br/>
<div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_1"><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="E1:49_3_1"/><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</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#R1">=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace4.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E2:49_3_1"/><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/><a NAME="E3:49_3_1"/><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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> <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/>;<br/><a NAME="E4:49_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E5:49_3_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</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> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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="txt" href="csspace4.html#E4"><i><font color="Green">Lemma42</font></i></a>, , , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_2"><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="E1:49_3_2"/><i><font color="Green">E89</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">h1</font>
 ;<br/><a NAME="E2:49_3_2"/>
<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">h1</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/>;<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="E4:49_3_2"/><i><font color="Green">E90</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">h1</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="E5:49_3_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:49_3"/><i><font color="Green">E91</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="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">h1</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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> ) implies  <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">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<a NAME="E5:49_3"/>
<span class="p1">(<span class="default"><a href="csspace4.html#K8" target="_self">ComplexBoundedFunctionsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <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">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="csspace4.html#E1:49"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E6:49_3"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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="txt" href="csspace4.html#E4"><i><font color="Green">Lemma42</font></i></a>, , <a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_3_3_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_3_3_1_1"><b>case </b></a><a NAME="E1:49_3_3_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/><div class="add"><div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_3_1_1_1"><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="E1:49_3_3_1_1_1"/>
<font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="ref" href="csspace4.html#D1" target="_self">Def1</a></i>;<br/><b>then </b><i><font color="Green">E106</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K12">"</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K12">"</a> </span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">c</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
;<br/><a NAME="E3:49_3_3_1_1_1"/><i><font color="Green">E107</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K12">"</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K12">"</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/><a NAME="E4:49_3_3_1_1_1"/><i><font color="Green">E108</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</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> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:49_3_3_1_1_1"/><i><font color="Green">E109</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K12">"</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K12">"</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <font color="Maroon">c</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T134">COMPLEX1:134</a>, <a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>
<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K2">"</a> 

;<br/><b>hence </b><a NAME="E7:49_3_3_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, <a class="txt" href="csspace4.html#E5"><i><font color="Green">Lemma48</font></i></a>, , , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E110</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_3_1_1_2"><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="E1:49_3_3_1_1_2"/><i><font color="Green">E111</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">f1</font>
 ;<br/><a NAME="E2:49_3_3_1_1_2"/>
<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">f1</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#E5"><i><font color="Green">Lemma48</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="E4:49_3_3_1_1_2"/><i><font color="Green">E112</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">f1</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="E5:49_3_3_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E4:49_3_3_1_1"/><i><font color="Green">E113</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><a NAME="E5:49_3_3_1_1"/><i><font color="Green">E114</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="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">f1</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> ) implies  <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">f1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> )
 <b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/><a NAME="E6:49_3_3_1_1"/>
<span class="p1">(<span class="default"><a href="csspace4.html#K8" target="_self">ComplexBoundedFunctionsNorm</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="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i><a class="txt" href="csspace4.html#E1:49"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E7:49_3_3_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<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> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></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/><a NAME="E8:49_3_3_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E9:49_3_3_1_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:49_3_3_1_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 ;<br/><a NAME="E11:49_3_3_1_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="csspace4.html#E5"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>hence </b><a NAME="E12:49_3_3_1_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_3_3_1_2"><b>case </b></a><a NAME="E1:49_3_3_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">fz</font> = <font color="Maroon">f</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K5" target="_self">C_VectorSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> ;<br/><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="csspace4.html#K4" target="_self">ComplexBoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K3">ComplexVectSpace</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">fz</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="csspace4.html#K5" target="_self">C_VectorSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="clvect_1.html#T2">CLVECT_1:2</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>

;<br/><b>hence </b><a NAME="E4:49_3_3_1_2"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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> <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">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, <a class="txt" href="csspace4.html#E2:49"><i><font color="Green">E34</font></i></a>, <a class="ref" href="complex1.html#T130">COMPLEX1:130</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:49_3"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:49"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <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> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">g</font></span><a href="clvect_1.html#K5">.||</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font>, <font color="Maroon">g1</font> = <font color="Maroon">g</font>, <font color="Maroon">h1</font> = <font color="Maroon">f</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">g</font> as  <a href="csspace4.html#V1" target="_self">bounded</a> <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> <b>by </b><i/>;<br/>
<div><i><font color="Green">E117</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4_1"><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="E1:49_4_1"/><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</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#R1">=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="csspace4.html#E2"><i><font color="Green">Lemma35</font></i></a></i>;<br/><a NAME="E2:49_4_1"/><i><font color="Green">E122</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/><a NAME="E3:49_4_1"/><i><font color="Green">E123</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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> <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/>;<br/><a NAME="E4:49_4_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</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> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">g</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:49_4_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</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> <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> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">g</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E6:49_4_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</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> <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> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">g</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="csspace4.html#E4"><i><font color="Green">Lemma42</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4_2"><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="E1:49_4_2"/><i><font color="Green">E125</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">h1</font>
 ;<br/><a NAME="E2:49_4_2"/>
<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">h1</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/>;<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="E4:49_4_2"/><i><font color="Green">E126</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">h1</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="E5:49_4_2"/>
<font color="Maroon">r</font> <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> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">g</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:49_4"/><i><font color="Green">E127</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="csspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">h1</font> holds <br/><font color="Olive">s</font> <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> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">g</font></span><a href="clvect_1.html#K5">.||</a></span> ) implies  <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">h1</font></span>)</span> <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> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">g</font></span><a href="clvect_1.html#K5">.||</a></span> )
 
<b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<a NAME="E5:49_4"/>
<span class="p1">(<span class="default"><a href="csspace4.html#K8" target="_self">ComplexBoundedFunctionsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <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">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="csspace4.html#E1:49"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:49_4"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <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> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">g</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="csspace4.html#E4"><i><font color="Green">Lemma42</font></i></a>, , <a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:49"/>
( ( <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> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> ) &amp; ( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> implies <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> <a href="hidden.html#R1">=</a> 0 ) &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</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> &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <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> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">g</font></span><a href="clvect_1.html#K5">.||</a></span> )
 <b>by </b><i>, , </i>;<br/>


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