<?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/>

<a NAME="E1:49"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span><br/> for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font></span>)</span> &amp; <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> st <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">z</font></span>)</span> &amp; <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font></span>)</span> &amp; <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">y</font></span>)</span> &amp; <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K3">+</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> &amp; 1 <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">RBLOP</font> =  <a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">BLOP</font> =  <a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">BS</font> =  <a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">ADD</font> =  <a href="rsspace.html#K8">Add_</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">MULT</font> =  <a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">ZE</font> =  <a href="rsspace.html#K10">Zero_</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">EXMULT</font> =  <a href="rsspace.html#K9">Mult_</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">UNIT</font> =  <a href="lopban_2.html#K7" target="_self">FuncUnit</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">NRM</font> =  <a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<a NAME="E2:49"/><i><font color="Green">E46</font></i>: 
( ( for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> holds  <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</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_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K3">+</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</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_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> ) &amp; ( 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_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> holds  1 <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> ;<br/><i><font color="Green">E47</font></i>: <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p3">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</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">x1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
;<br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x1</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">y1</font></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">y1</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<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> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="normsp_1.html#U1">norm</a> of <span class="p2">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<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> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">y</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
;<br/><b>hence </b><a NAME="E4:49_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</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">x</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">y</font></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/><i><font color="Green">E48</font></i>: <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x1</font></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<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="normsp_1.html#D1">NORMSP_1:def 1</a></i>
;<br/> <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> = 
 <a href="rsspace.html#K10">Zero_</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E7:49_1"/>
( <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> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/><b>thus </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">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</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">x</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p3">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</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">x1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</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#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="normsp_1.html#D2">NORMSP_1:def 2</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49"/>
(  <a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="normsp_1.html#V2">RealNormSpace-like</a> &amp;  <a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V6">right_complementable</a> &amp;  <a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="group_1.html#V4">associative</a> &amp;  <a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="funcsdom.html#V2">Algebra-like</a> &amp;  <a href="lopban_2.html#K10" target="_self">R_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V7">RealLinearSpace-like</a> )
 <b>by </b><i>, , <a class="ref" href="funcsdom.html#D20">FUNCSDOM:def 20</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a>, <a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a>, <a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>


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