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

<b>set </b><font color="Maroon">RBLOP</font> =  <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">BLOP</font> =  <a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">BS</font> =  <a href="clopban1.html#K14">C_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="csspace.html#K8">Add_</a> <span class="p1">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="clopban1.html#K6">C_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="clopban2.html#K5" target="_self">FuncMult</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">ZE</font> =  <a href="csspace.html#K10">Zero_</a> <span class="p1">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="clopban1.html#K6">C_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="csspace.html#K9">Mult_</a> <span class="p1">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="clopban1.html#K6">C_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="clopban2.html#K6" target="_self">FuncUnit</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">NRM</font> =  <a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>thus </b><a NAME="E1:55_1_1"/>
 <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="clopban2.html#V2" target="_self">Banach_Algebra-like_1</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="clopban2.html#D12" target="_self">CLOPBAN2:def 12</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>proof </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="clopban2.html#K9" target="_self">C_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="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="clopban2.html#D9" target="_self">CLOPBAN2:def 9</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="subset_1.html#M2">Element</a> of  <a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> ;<br/>
<i><font color="Green">E42</font></i>: <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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="clopban2.html#K5" target="_self">FuncMult</a> <font color="Maroon">X</font></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="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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="clopban2.html#K3" target="_self">*</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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="clopban1.html#K11">modetrans</a> <font color="Maroon">x1</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">y1</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span>

;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">x1</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">y1</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13">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#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">y1</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban1.html#D10">CLOPBAN1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13">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#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13">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="clopban1.html#D10">CLOPBAN1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<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="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">y</font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E4:55_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">y</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:55_1_1"/>
 <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="clopban2.html#V3" target="_self">Banach_Algebra-like_2</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:55_1_1_2"/>
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clopban2.html#K6" target="_self">FuncUnit</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p3">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
1
<b>by </b><i/>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="clopban2.html#D10" target="_self">CLOPBAN2:def 10</a><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:55_1_1"/>
 <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="clopban2.html#V4" target="_self">Banach_Algebra-like_3</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="cfunct_1.html#NM1">Complex</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="clopban2.html#D11" target="_self">CLOPBAN2:def 11</a><br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>;<br/>



<b>thus </b><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">a</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:55_1_1"/>
 <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="vectsp_1.html#V8">left_unital</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a><br/>

<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="clopban2.html#K5" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="clopban2.html#K6" target="_self">FuncUnit</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban2.html#K6" target="_self">FuncUnit</a> <font color="Maroon">X</font></span>)</span> <a href="clopban2.html#K3" target="_self">*</a> <font color="Maroon">xx</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:55_1_1"/>
 <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="vectsp_1.html#V5">left-distributive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D12">VECTSP_1:def 12</a><br/>

<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font>, <font color="Maroon">yy</font> = <font color="Maroon">y</font>, <font color="Maroon">zz</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="clopban2.html#K5" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K8">Add_</a> <span class="p3">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="clopban1.html#K6">C_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">y</font>,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="clopban2.html#K2" target="_self">+</a> <font color="Maroon">zz</font></span>)</span> <a href="clopban2.html#K3" target="_self">*</a> <font color="Maroon">xx</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="clopban2.html#K3" target="_self">*</a> <font color="Maroon">xx</font></span>)</span> <a href="clopban2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">zz</font> <a href="clopban2.html#K3" target="_self">*</a> <font color="Maroon">xx</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K8">Add_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="clopban2.html#K3" target="_self">*</a> <font color="Maroon">xx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban2.html#K5" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">zz</font>,<font color="Maroon">xx</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K8">Add_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K6">C_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> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban2.html#K5" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:55_1_1_6"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">seq</font> is <a href="csspace3.html#V1">CCauchy</a>
 ;<br/><b>reconsider </b><font color="Maroon">seq1</font> = <font color="Maroon">seq</font> as   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_6_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="E1:55_1_1_6_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:55_1_1_6_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="csspace3.html#T10">CSSPACE3:10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_6_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> )
 ;<br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#T4">CLVECT_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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="csspace.html#K8">Add_</a> <span class="p3">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="clopban1.html#K6">C_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> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></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="clopban1.html#K13">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="csspace.html#K8">Add_</a> <span class="p3">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="clopban1.html#K6">C_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> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p4">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p4">(<span class="default"><a href="clopban1.html#K6">C_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="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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="csspace.html#K8">Add_</a> <span class="p3">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="clopban1.html#K6">C_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> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></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="clopban1.html#K13">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"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#T4">CLVECT_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
;<br/><b>hence </b><a NAME="E3:55_1_1_6_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:55_1_1_6_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div><a NAME="E4:55_1_1_6"/><b>then </b>
<font color="Maroon">seq1</font> is <a href="csspace3.html#V1">CCauchy</a>
 <b>by </b><i><a class="ref" href="csspace3.html#T10">CSSPACE3:10</a></i>;<br/><a NAME="E5:55_1_1_6"/><b>then </b>
<font color="Maroon">seq1</font> is <a href="clvect_1.html#V6">convergent</a>
 <b>by </b><i><a class="ref" href="clopban1.html#D14">CLOPBAN1:def 14</a></i>;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E7:55_1_1_6"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g1</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D15">CLVECT_1:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_6_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="E1:55_1_1_6_2"/><i><font color="Green">E56</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:55_1_1_6_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g1</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_6_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_6_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g1</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g1</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">g1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">g1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#T4">CLVECT_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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="csspace.html#K8">Add_</a> <span class="p3">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="clopban1.html#K6">C_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> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">g1</font></span>)</span></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="clopban1.html#K13">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="csspace.html#K8">Add_</a> <span class="p3">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="clopban1.html#K6">C_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> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p4">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p4">(<span class="default"><a href="clopban1.html#K6">C_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="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span>,<font color="Maroon">g1</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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="csspace.html#K8">Add_</a> <span class="p3">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="clopban1.html#K6">C_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> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">g</font></span>)</span></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="clopban1.html#K13">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"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#T4">CLVECT_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">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"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
;<br/><b>hence </b><a NAME="E3:55_1_1_6_2_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:55_1_1_6_2"/>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:55_1_1_6"/>
<font color="Maroon">seq</font> is <a href="clvect_1.html#V6">convergent</a>
 <b>by </b><i><a class="ref" href="clvect_1.html#D15">CLVECT_1:def 15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:55_1_1"/>
 <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">X</font> is <a href="clopban1.html#V4">complete</a>
 <b>by </b><i><a class="ref" href="clopban1.html#D14">CLOPBAN1:def 14</a></i>;<br/>


</div>
