<?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">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K7" title="LOPBAN_2:func.7">Ring_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>



<b>set </b><font color="Maroon" title="c5">RBLOP</font> =  <a href="lopban_2.html#K7" title="LOPBAN_2:func.7">Ring_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c6">BLOP</font> =  <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>set </b><font color="Maroon" title="c7">ADD</font> =  <a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</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">MULT</font> =  <a href="lopban_2.html#K5" title="LOPBAN_2:func.5">FuncMult</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c9">ZE</font> =  <a href="rsspace.html#K10" title="RSSPACE:func.10">Zero_</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="c10">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="c11">UNIT</font> =  <a href="lopban_2.html#K6" title="LOPBAN_2:func.6">FuncUnit</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c12">RRL</font> =  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</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="rsspace.html#K10" title="RSSPACE:func.10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</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>,<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> #);<br/>
<b>set </b><font color="Maroon" title="c13">IT</font> =  <a href="lopban_2.html#K7" title="LOPBAN_2:func.7">Ring_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c16">h</font> = <font color="Maroon" title="c4">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</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="rsspace.html#K10" title="RSSPACE:func.10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</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>,<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> #) ;<br/>
<b>thus </b><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c3">y</font> = 
<font color="Maroon" title="c14">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">g</font>

<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">h</font>

<br/>.= 
<font color="Maroon" title="c14">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">h</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">z</font></span>)</span>

;<br/>

<b>thus </b><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="lopban_2.html#K7" title="LOPBAN_2:func.7">Ring_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c14">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</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="rsspace.html#K10" title="RSSPACE:func.10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</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>,<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> #)</span>)</span>

<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/>

<b>thus </b><a NAME="E5:24"/>
 ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K7" title="LOPBAN_2:func.7">Ring_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lopban_2.html#K7" title="LOPBAN_2:func.7">Ring_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c17">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</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="rsspace.html#K10" title="RSSPACE:func.10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</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>,<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> #)<b> such that </b><br/><a NAME="E2:24_4"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c14">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</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="rsspace.html#K10" title="RSSPACE:func.10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</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>,<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> #)
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">t</font> = <font color="Maroon" title="c17">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K7" title="LOPBAN_2:func.7">Ring_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c18">t</font>
;<br/>

<b>thus </b><a NAME="E4:24_4"/>
<font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lopban_2.html#K7" title="LOPBAN_2:func.7">Ring_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="lopban_2.html#E2:24_4"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>


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

<b>reconsider </b><font color="Maroon" title="c17">xx</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c18">yy</font> = <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c19">zz</font> = <font color="Maroon" title="c4">z</font> as    <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>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">z</font> = 
<span class="p1">(<span class="default"><a href="lopban_2.html#K5" title="LOPBAN_2:func.5">FuncMult</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">xx</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c18">yy</font></span>)</span>,<font color="Maroon" title="c19">zz</font>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">xx</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c18">yy</font></span>)</span> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c19">zz</font>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c17">xx</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">yy</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c19">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#T7" target="_self" title="LOPBAN_2:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_2.html#K5" title="LOPBAN_2:func.5">FuncMult</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">xx</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">yy</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c19">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="lopban_2.html#K7" title="LOPBAN_2:func.7">Ring_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c17">xx</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban_2.html#K6" title="LOPBAN_2:func.6">FuncUnit</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="lopban_2.html#T8" target="_self" title="LOPBAN_2:th.8">Th8</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="lopban_2.html#K7" title="LOPBAN_2:func.7">Ring_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><a href="lopban_2.html#K6" title="LOPBAN_2:func.6">FuncUnit</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c17">xx</font>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="lopban_2.html#T8" target="_self" title="LOPBAN_2:th.8">Th8</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">z</font></span>)</span> = 
<font color="Maroon" title="c17">xx</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">yy</font> <a href="lopban_2.html#K2" title="LOPBAN_2:func.2">+</a> <font color="Maroon" title="c19">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">xx</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c18">yy</font></span>)</span> <a href="lopban_2.html#K2" title="LOPBAN_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">xx</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c19">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#T9" target="_self" title="LOPBAN_2:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</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#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">xx</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c18">yy</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_2.html#K5" title="LOPBAN_2:func.5">FuncMult</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">xx</font>,<font color="Maroon" title="c19">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">yy</font> <a href="lopban_2.html#K2" title="LOPBAN_2:func.2">+</a> <font color="Maroon" title="c19">zz</font></span>)</span> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c17">xx</font>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">yy</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c17">xx</font></span>)</span> <a href="lopban_2.html#K2" title="LOPBAN_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">zz</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c17">xx</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#T10" target="_self" title="LOPBAN_2:th.10">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</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#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">yy</font> <a href="lopban_2.html#K3" title="LOPBAN_2:func.3">*</a> <font color="Maroon" title="c17">xx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_2.html#K5" title="LOPBAN_2:func.5">FuncMult</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c19">zz</font>,<font color="Maroon" title="c17">xx</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_2.html#D4" target="_self" title="LOPBAN_2:def.4">Def4</a></i>
;<br/>


</div>
