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

<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="normsp_1.html#NM1">RealNormSpace</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="lopban_1.html#K16" target="_self">R_NormSpace_of_BoundedLinearOperators</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="lopban_1.html#K16" target="_self">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</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="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">z</font> = the <a href="struct_0.html#U1">carrier</a> of <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> = the <a href="struct_0.html#U1">carrier</a> of <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 the <a href="struct_0.html#U1">carrier</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="74_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:74_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E2:74_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="lopban_1.html#E2:74"><i><font color="Green">E52</font></i></a>, <a class="ref" href="normsp_1.html#T8">NORMSP_1:8</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="normsp_1.html#D2">NORMSP_1:def 2</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="E5:74_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="E6:74_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="lopban_1.html#K16" target="_self">R_NormSpace_of_BoundedLinearOperators</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">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_2"/><i><font color="Green">E53</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="lopban_1.html#K16" target="_self">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:74_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">z</font> = the <a href="struct_0.html#U1">carrier</a> of <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> = the <a href="struct_0.html#U1">carrier</a> of <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 the <a href="struct_0.html#U1">carrier</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="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E3:74_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">z</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E1:74_2"><i><font color="Green">E53</font></i></a>, </i>;<br/>
<div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_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="E1:74_2_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:74_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="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">t</font></span><a href="normsp_1.html#K1">.||</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="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:74_2_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="lopban_1.html#E3:74_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E6:74_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><a class="ref" href="lopban_1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:74_2_1"/><i><font color="Green">E91</font></i>: 
( not  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="lopban_1.html#K14" 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="E7:74_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14" 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="ref" href="lopban_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E9:74_2_1"/><i><font color="Green">E93</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T4" target="_self">Th4</a>, <a class="ref" href="lopban_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E10:74_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="lopban_1.html#K14" 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="lopban_1.html#K14" 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="E11:74_2_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="lopban_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="lopban_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E12:74_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lopban_1.html#K15" 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> 0
 
<b>by </b><i><a class="ref" href="lopban_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E13:74_2_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:74"/><i><font color="Green">E94</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_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">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font> as  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.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">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:74_3_1"/><i><font color="Green">E131</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><a NAME="E2:74_3_1"/><i><font color="Green">E132</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</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="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:74_3_1"/><i><font color="Green">E148</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</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="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/><a NAME="E4:74_3_1"/><i><font color="Green">E149</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:74_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E6:74_3_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="lopban_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E7:74_3_1"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:74_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E9:74_3_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="lopban_1.html#T5" target="_self">Th5</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">E151</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_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="E1:74_3_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">h1</font>
 ;<br/><a NAME="E2:74_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="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">t</font></span><a href="normsp_1.html#K1">.||</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="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:74_3_2"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E5:74_3_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="lopban_1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:74_3"/><i><font color="Green">E154</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="lopban_1.html#K14" 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">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> ) implies  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> )
 
<b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<a NAME="E5:74_3"/>
<span class="p1">(<span class="default"><a href="lopban_1.html#K15" target="_self">BoundedLinearOperatorsNorm</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">a</font> <a href="rlvect_1.html#K3">*</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="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:74_3"/><b>then </b><i><font color="Green">E155</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_3_3_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_3_3_1_1"><b>case </b></a><a NAME="E1:74_3_3_1_1"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><i><font color="Green">E157</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3_3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:74_3_3_1_1_1"/><i><font color="Green">E158</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><a NAME="E2:74_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">a</font> <a href="rlvect_1.html#K3">*</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/>;<br/><b>then </b><i><font color="Green">E159</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</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">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</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="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
1 <a href="rlvect_1.html#K3">*</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="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="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><a NAME="E4:74_3_3_1_1_1"/><i><font color="Green">E160</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</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="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/><a NAME="E5:74_3_3_1_1_1"/><i><font color="Green">E161</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:74_3_3_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E7:74_3_3_1_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E8:74_3_3_1_1_1"/><b>then </b><i><font color="Green">E162</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T8" target="_self">Th8</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:74_3_3_1_1_1"/><i><font color="Green">E163</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T15">ABSVALUE:15</a></i>
<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</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">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K2">"</a> 

;<br/><b>hence </b><a NAME="E11:74_3_3_1_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T6" target="_self">Th6</a>, <a class="ref" href="lopban_1.html#T7" target="_self">Th7</a>, <a class="ref" href="lopban_1.html#T9" target="_self">Th9</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">E164</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_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="E1:74_3_3_1_1_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">f1</font>
 ;<br/><a NAME="E2:74_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="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="lopban_1.html#T6" target="_self">Th6</a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:74_3_3_1_1_2"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E5:74_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">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="lopban_1.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><a NAME="E4:74_3_3_1_1"/><i><font color="Green">E167</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> <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="E5:74_3_3_1_1"/><i><font color="Green">E168</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="lopban_1.html#K14" 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">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> ) implies  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" 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">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> )
 <b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/><a NAME="E6:74_3_3_1_1"/>
<span class="p1">(<span class="default"><a href="lopban_1.html#K15" 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="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E7:74_3_3_1_1"/><b>then </b><i><font color="Green">E169</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="lopban_1.html#T7" target="_self">Th7</a>, <a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>;<br/><a NAME="E8:74_3_3_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E9:74_3_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:74_3_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 ;<br/><a NAME="E11:74_3_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>hence </b><a NAME="E12:74_3_3_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_3_3_1_2"><b>case </b></a><a NAME="E1:74_3_3_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">fz</font> = <font color="Maroon">f</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K11" target="_self">R_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> ;<br/><i><font color="Green">E172</font></i>: <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font> = 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">fz</font>

<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K11" target="_self">R_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K16" target="_self">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>

;<br/><b>thus </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> = 
0 <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:74_3"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:74"/>
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_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="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.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">E173</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:74_4_1"/><i><font color="Green">E174</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><a NAME="E2:74_4_1"/><i><font color="Green">E175</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="txt" href="lopban_1.html#E6"><i><font color="Green">Lemma48</font></i></a></i>;<br/><a NAME="E3:74_4_1"/><i><font color="Green">E176</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/><a NAME="E4:74_4_1"/><i><font color="Green">E177</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:74_4_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:74_4_1"/><b>then </b><i><font color="Green">E178</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E7:74_4_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E8:74_4_1"/><b>then </b><i><font color="Green">E179</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="lopban_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E9:74_4_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E10:74_4_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="lopban_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E11:74_4_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E12:74_4_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E13:74_4_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="lopban_1.html#T5" target="_self">Th5</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">E180</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_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="E1:74_4_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">h1</font>
 ;<br/><a NAME="E2:74_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="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">t</font></span><a href="normsp_1.html#K1">.||</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="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:74_4_2"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E5:74_4_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="lopban_1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:74_4"/><i><font color="Green">E183</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="lopban_1.html#K14" 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="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span> ) implies  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span> )
 
<b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<a NAME="E5:74_4"/>
<span class="p1">(<span class="default"><a href="lopban_1.html#K15" target="_self">BoundedLinearOperatorsNorm</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="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:74_4"/>
<span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:74"/>
( ( <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</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="lopban_1.html#K16" target="_self">R_NormSpace_of_BoundedLinearOperators</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="lopban_1.html#K16" target="_self">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> implies <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0 ) &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">g</font></span><a href="normsp_1.html#K1">.||</a></span> )
 <b>by </b><i><a class="ref" href="lopban_1.html#T3" target="_self">Th3</a>, <a class="txt" href="lopban_1.html#E2:74"><i><font color="Green">E52</font></i></a>, </i>;<br/>


</div>
