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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">f</font> <a href="clopban2.html#K3" title="CLOPBAN2:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Olive" title="b3">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="clopban2.html#K3" title="CLOPBAN2:func.3">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="clopban2.html#K3" title="CLOPBAN2:func.3">*</a> <font color="Olive" title="b3">h</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">h</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="clopban2.html#K3" title="CLOPBAN2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="clopban2.html#K3" title="CLOPBAN2:func.3">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="clopban2.html#K3" title="CLOPBAN2:func.3">*</a> <font color="Maroon" title="c4">h</font></span>)</span></span><br/>



<b>set </b><font color="Maroon" title="c5">BLOP</font> =  <a href="clopban1.html#K14" title="CLOPBAN1:func.14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c6">ADD</font> =  <a href="csspace.html#K8" title="CSSPACE:func.8">Add_</a> <span class="p1">(<span class="default"><a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<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="c1">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">mf</font> =  <a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c8">mg</font> =  <a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c9">mh</font> =  <a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c10">mgh</font> =  <a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:15"/>
<span class="p1">(<span class="default"><a href="csspace.html#K8" title="CSSPACE:func.8">Add_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<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="c1">X</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">gg</font> = <font color="Maroon" title="c3">g</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14" title="CLOPBAN1:func.14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">hh</font> = <font color="Maroon" title="c4">h</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14" title="CLOPBAN1:func.14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<a NAME="E3:15_1"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c11">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="clopban1.html#D10" title="CLOPBAN1:def.10">CLOPBAN1:def 10</a></i>;<br/>
<a NAME="E4:15_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c12">hh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="clopban1.html#D10" title="CLOPBAN1:def.10">CLOPBAN1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">k</font> = <span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14" title="CLOPBAN1:func.14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="clopban1.html#D8" title="CLOPBAN1:def.8">CLOPBAN1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">fg</font> = <span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14" title="CLOPBAN1:func.14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="clopban1.html#D8" title="CLOPBAN1:def.8">CLOPBAN1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">fh</font> = <span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14" title="CLOPBAN1:func.14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="clopban1.html#D8" title="CLOPBAN1:def.8">CLOPBAN1:def 8</a></i>;<br/>
<a NAME="E8:15_1"/>
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">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span></span><br/>

<a NAME="E1:15_1_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">gg</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">hh</font>
 
;<br/>
<a NAME="E2:15_1_1"/>
 <a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font>
 
<b>by </b><i><a class="ref" href="clopban1.html#D10" title="CLOPBAN1:def.10">CLOPBAN1:def 10</a></i>;<br/>
<a NAME="E3:15_1_1"/><b>then </b><i><font color="Green" title="E14">A4</font></i>: 
<span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:15_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E4:15_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="clopban1.html#T39" title="CLOPBAN1:th.39">CLOPBAN1:39</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font> = 
<span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban2.html#T4" target="_self" title="CLOPBAN2:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:15_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="clopban1.html#D3" title="CLOPBAN1:def.3">CLOPBAN1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban2.html#T4" target="_self" title="CLOPBAN2:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p3">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban2.html#T4" target="_self" title="CLOPBAN2:th.4">Th4</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:15_1"/><b>then </b>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">fg</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">fh</font>
 
<b>by </b><i><a class="ref" href="clopban1.html#T39" title="CLOPBAN1:th.39">CLOPBAN1:39</a></i>;<br/>
<b>hence </b><a NAME="E10:15_1"/>
<span class="p1">(<span class="default"><a href="csspace.html#K8" title="CSSPACE:func.8">Add_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<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="c1">X</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="clopban2.html#K1" title="CLOPBAN2:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11" title="CLOPBAN1:func.11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:15"/>
<font color="Maroon" title="c2">f</font> <a href="clopban2.html#K3" title="CLOPBAN2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="clopban2.html#K3" title="CLOPBAN2:func.3">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="clopban2.html#K2" title="CLOPBAN2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="clopban2.html#K3" title="CLOPBAN2:func.3">*</a> <font color="Maroon" title="c4">h</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
