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

<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>, <font color="Maroon">Y</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/>

<b>set </b><font color="Maroon">W</font> =  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>
<a NAME="E1:47"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K8" target="_self">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K8" target="_self">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K8" target="_self">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> as   <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="E3:47_1"/>
<font color="Maroon">f</font> is <a href="lopban_1.html#V3" target="_self">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</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><a class="ref" href="lopban_1.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</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><a class="ref" href="lopban_1.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>consider </b><font color="Maroon">K1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:47_1_1"/><i><font color="Green">E52</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K1</font> &amp; ( for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K1</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">K2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:47_1_1"/><i><font color="Green">E53</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K2</font> &amp; ( for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K2</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> ) )
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">K3</font> = <font color="Maroon">K1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">K2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lopban_1.html#D9" target="_self">LOPBAN_1:def 9</a><br/>

<a NAME="E7:47_1_1"/><i><font color="Green">E54</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">K2</font>
 
<b>by </b><i>, <a class="ref" href="lopban_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:47_1_1_1"/><i><font color="Green">E86</font></i>: 
<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="funct_2.html#K3">.</a> <font color="Maroon">x</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">u1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E2:47_1_1_1"/><i><font color="Green">E89</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">u1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">u1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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="E3:47_1_1_1"/><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K1</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E4:47_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">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K2</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E5:47_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">u1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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"><font color="Maroon">K1</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</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"><font color="Maroon">K2</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</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/><b>hence </b><a NAME="E6:47_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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K3</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:47_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K3</font> &amp; ( for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <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="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K3</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47"/>
for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K8" target="_self">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> holds <br/><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K8" target="_self">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:47_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> as   <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="E3:47_2"/>
<font color="Maroon">f</font> is <a href="lopban_1.html#V3" target="_self">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</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><a class="ref" href="lopban_1.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>consider </b><font color="Maroon">K</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:47_2_1"/><i><font color="Green">E92</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> &amp; ( for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> ) )
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">K1</font> = <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> <font color="Maroon">K</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lopban_1.html#D9" target="_self">LOPBAN_1:def 9</a><br/>

<a NAME="E4:47_2_1"/><i><font color="Green">E93</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> &amp; 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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:47_2_1_1"/><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> <span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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="E2:47_2_1_1"/><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:47_2_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="E4:47_2_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"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E5:47_2_1_1"/>
<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="funct_2.html#K3">.</a> <font color="Maroon">x</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#K4">*</a> <font color="Maroon">K</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">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:47_2_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K1</font> &amp; ( for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <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="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K1</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> ) )
 <b>by </b><i><a class="ref" href="lopban_1.html#T2" target="_self">Th2</a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_2"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:47"/>
 <a href="lopban_1.html#K10" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a></i>;<br/>


</div>
