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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">W</font> =  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>;<br/>
<a NAME="E1:43"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">u</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 <span class="kw">and </span><br/><a NAME="E2:43_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> as   <a href="clopban1.html#NM1" title="CLOPBAN1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#D5" target="_self" title="CLOPBAN1:def.5">Def5</a></span>;<br/>
<a NAME="E4:43_1"/>
<font color="Maroon" title="c6">f</font> is  <a href="clopban1.html#V2" title="CLOPBAN1:attr.2">bounded</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="clopban1.html#V2" title="CLOPBAN1:attr.2">bounded</a>  <a href="clopban1.html#NM1" title="CLOPBAN1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="clopban1.html#D8" target="_self" title="CLOPBAN1:def.8">Def8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">K2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:43_1_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">K2</font>
 <span class="kw">and </span><br/><a NAME="E4:43_1_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">K2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#D7" target="_self" title="CLOPBAN1:def.7">Def7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">u1</font> = <font color="Maroon" title="c5">u</font> as   <a href="clopban1.html#V2" title="CLOPBAN1:attr.2">bounded</a>  <a href="clopban1.html#NM1" title="CLOPBAN1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="clopban1.html#D8" target="_self" title="CLOPBAN1:def.8">Def8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">K1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E7:43_1_1"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K1</font>
 <span class="kw">and </span><br/><a NAME="E8:43_1_1"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#D7" target="_self" title="CLOPBAN1:def.7">Def7</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c11">K3</font> = <font color="Maroon" title="c10">K1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">K2</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="clopban1.html#D7" target="_self" title="CLOPBAN1:def.7">CLOPBAN1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">K3</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">K3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">K3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span></span><br/><a NAME="E1:43_1_1_1"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D14" title="CLVECT_1:def.14">CLVECT_1:def 14</a></span>;<br/><a NAME="E2:43_1_1_1"/><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">K2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a></span>;<br/><a NAME="E3:43_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a></span>;<br/><a NAME="E4:43_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A10</font></span>: 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">K1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">K2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E5:43_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#T18" target="_self" title="CLOPBAN1:th.18">Th18</a></span>;<br/><span class="kw">hence </span><a NAME="E6:43_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">K3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="txt" href="#E4:43_1_1_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:43_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">K3</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">K3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_1_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E3:43_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:43_1"/>
<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#D8" target="_self" title="CLOPBAN1:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:43"/>
for <font color="Olive" title="b1">c</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b2">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b1">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">c</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon" title="c4">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c4">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:43_2"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c4">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> as   <a href="clopban1.html#NM1" title="CLOPBAN1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#D5" target="_self" title="CLOPBAN1:def.5">Def5</a></span>;<br/>
<a NAME="E3:43_2"/>
<font color="Maroon" title="c6">f</font> is  <a href="clopban1.html#V2" title="CLOPBAN1:attr.2">bounded</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c5">v</font> as   <a href="clopban1.html#V2" title="CLOPBAN1:attr.2">bounded</a>  <a href="clopban1.html#NM1" title="CLOPBAN1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="clopban1.html#D8" target="_self" title="CLOPBAN1:def.8">Def8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">K</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:43_2_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">K</font>
 <span class="kw">and </span><br/><a NAME="E4:43_2_1"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">K</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#D7" target="_self" title="CLOPBAN1:def.7">Def7</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c9">K1</font> = <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">K</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="clopban1.html#D7" target="_self" title="CLOPBAN1:def.7">CLOPBAN1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">K1</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">K1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> ) )</span><br/>

<div><span class="lab"><font color="Green" title="E31">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">K</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span></span><br/><a NAME="E1:43_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/><a NAME="E2:43_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">K</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_2_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E3:43_2_1_1"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D14" title="CLVECT_1:def.14">CLVECT_1:def 14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:43_2_1_1"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">K</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2_1_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="clopban1.html#T19" target="_self" title="CLOPBAN1:th.19">Th19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:43_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:43_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">K1</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">K1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E5:43_2_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:43_2"/>
<font color="Maroon" title="c4">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#D8" target="_self" title="CLOPBAN1:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:43"/>
 <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="clvect_1.html#D7" title="CLVECT_1:def.7">CLVECT_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
