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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>set </b><font color="Maroon" title="c6">BLOP</font> =  <a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_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="c7">EXMULT</font> =  <a href="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K10" title="LOPBAN_1:func.10">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="lopban_1.html#K8" title="LOPBAN_1:func.8">R_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="c8">mf</font> =  <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="c9">mg</font> =  <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="c10">maf</font> =  <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c11">mbg</font> =  <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c12">mfg</font> =  <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:17"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10" title="LOPBAN_1:func.10">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="lopban_1.html#K8" title="LOPBAN_1:func.8">R_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"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="lopban_2.html#K1" title="LOPBAN_2:func.1">*</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lopban_2.html#K1" title="LOPBAN_2:func.1">*</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</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="17_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c13">ff</font> = <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c14">gg</font> = <font color="Maroon" title="c3">g</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<a NAME="E2:17_1"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c13">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D12" title="LOPBAN_1:def.12">LOPBAN_1:def 12</a></i>;<br/>
<a NAME="E3:17_1"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c14">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="lopban_1.html#D12" title="LOPBAN_1:def.12">LOPBAN_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">fg</font> = <span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="lopban_2.html#K1" title="LOPBAN_2:func.1">*</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_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="lopban_1.html#D10" title="LOPBAN_1:def.10">LOPBAN_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">k</font> = <span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lopban_2.html#K1" title="LOPBAN_2:func.1">*</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_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="lopban_1.html#D10" title="LOPBAN_1:def.10">LOPBAN_1:def 10</a></i>;<br/>
<a NAME="E6:17_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_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="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lopban_2.html#K1" title="LOPBAN_2:func.1">*</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</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"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="lopban_2.html#K1" title="LOPBAN_2:func.1">*</a> <span class="p3">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>set </b><font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="c17">x</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">af</font> =  <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_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="lopban_1.html#D10" title="LOPBAN_1:def.10">LOPBAN_1:def 10</a></i>;<br/>
<a NAME="E2:17_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c13">ff</font>
 
;<br/>
<a NAME="E3:17_1_1"/>
 <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</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="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D12" title="LOPBAN_1:def.12">LOPBAN_1:def 12</a></i>;<br/>
<a NAME="E4:17_1_1"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="c17">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="c17">x</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="lopban_2.html#E2:17_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="lopban_2.html#E2:17_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="lopban_1.html#T42" title="LOPBAN_1:th.42">LOPBAN_1:42</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">bg</font> =  <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_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="lopban_1.html#D10" title="LOPBAN_1:def.10">LOPBAN_1:def 10</a></i>;<br/>
<a NAME="E6:17_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">gg</font>
 
;<br/>
<a NAME="E7:17_1_1"/>
 <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</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="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D12" title="LOPBAN_1:def.12">LOPBAN_1:def 12</a></i>;<br/>
<a NAME="E8:17_1_1"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</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="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="c17">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lopban_2.html#E3:17_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="lopban_2.html#E6:17_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="lopban_1.html#T42" title="LOPBAN_1:th.42">LOPBAN_1:42</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lopban_2.html#K1" title="LOPBAN_2:func.1">*</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</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="c17">x</font> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</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="c17">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#T4" target="_self" title="LOPBAN_2:th.4">Th4</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="c17">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_2.html#E4:17_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="lopban_2.html#E8:17_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="lopban_1.html#D6" title="LOPBAN_1:def.6">LOPBAN_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="lopban_1.html#K13" title="LOPBAN_1:func.13">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="c17">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="lopban_2.html#K1" title="LOPBAN_2:func.1">*</a> <span class="p3">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="c17">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#T4" target="_self" title="LOPBAN_2:th.4">Th4</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:17_1"/><b>then </b>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c15">fg</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T42" title="LOPBAN_1:th.42">LOPBAN_1:42</a></i>;<br/>
<b>hence </b><a NAME="E8:17_1"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10" title="LOPBAN_1:func.10">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="lopban_1.html#K8" title="LOPBAN_1:func.8">R_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"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="lopban_2.html#K1" title="LOPBAN_2:func.1">*</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lopban_2.html#K1" title="LOPBAN_2:func.1">*</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:17"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lopban_2.html#K4" title="LOPBAN_2:func.4">*</a> <font color="Maroon" title="c3">g</font></span>)</span>
 ;<br/>


</div>
