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

<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>set </b><font color="Maroon">BLOP</font> =  <a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">EXMULT</font> =  <a href="rsspace.html#K9">Mult_</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">mf</font> =  <a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">mg</font> =  <a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">maf</font> =  <a href="lopban_1.html#K13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">mbg</font> =  <a href="lopban_1.html#K13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">mfg</font> =  <a href="lopban_1.html#K13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<a NAME="E1:18"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font>, <font color="Maroon">gg</font> = <font color="Maroon">g</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> ;<br/>
<a NAME="E2:18_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">ff</font> <a href="hidden.html#R1">=</a>  <a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>;<br/>
<a NAME="E3:18_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">gg</font> <a href="hidden.html#R1">=</a>  <a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fg</font> = <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="lopban_1.html#D10">LOPBAN_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="lopban_1.html#D10">LOPBAN_1:def 10</a></i>;<br/>
<a NAME="E6:18_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">fg</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>;<br/>
<a NAME="E7:18_1"/>
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</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">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/>

<b>set </b><font color="Maroon">y</font> = <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">af</font> =  <a href="lopban_1.html#K13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="lopban_1.html#D10">LOPBAN_1:def 10</a></i>;<br/>
<a NAME="E2:18_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">ff</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>;<br/>
<a NAME="E3:18_1_1"/>
 <a href="lopban_1.html#K13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>;<br/>
<a NAME="E4:18_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="lopban_1.html#T42">LOPBAN_1:42</a></i>;<br/>
<b>reconsider </b><font color="Maroon">bg</font> =  <a href="lopban_1.html#K13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="lopban_1.html#D10">LOPBAN_1:def 10</a></i>;<br/>
<a NAME="E6:18_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">gg</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>;<br/>
<a NAME="E7:18_1_1"/>
 <a href="lopban_1.html#K13">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>;<br/>
<a NAME="E8:18_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="lopban_1.html#T42">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">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="lopban_1.html#D6">LOPBAN_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</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">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:18_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">fg</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#T42">LOPBAN_1:42</a></i>;<br/>
<b>hence </b><a NAME="E9:18_1"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:18"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="lopban_2.html#K5" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="lopban_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lopban_2.html#K5" target="_self">*</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
