<?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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>;<br/>



<b>set </b><font color="Maroon">RBLOP</font> =  <a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">BLOP</font> =  <a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">ADD</font> =  <a href="rsspace.html#K8">Add_</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">MULT</font> =  <a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">ZE</font> =  <a href="rsspace.html#K10">Zero_</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">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">UNIT</font> =  <a href="lopban_2.html#K7" target="_self">FuncUnit</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">RRL</font> =  <a href="rlvect_1.html#G2">RLSStruct</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="rsspace.html#K10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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>,<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> #);<br/>
<b>set </b><font color="Maroon">IT</font> =  <a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font>, <font color="Maroon">g</font> = <font color="Maroon">y</font>, <font color="Maroon">h</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G2">RLSStruct</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="rsspace.html#K10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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>,<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> #) ;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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> <font color="Maroon">f</font>,<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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> <font color="Maroon">g</font>,<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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"><span class="p2">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p3">(<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="p3">(<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> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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">f</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p3">(<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="p3">(<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> <font color="Maroon">g</font>,<font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="rsspace.html#K10">Zero_</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>
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</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="rsspace.html#K10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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>,<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> #)</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</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="rsspace.html#K10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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>,<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> #)</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/>

<b>thus </b><a NAME="E5:25"/>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> st <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G2">RLSStruct</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="rsspace.html#K10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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>,<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> #)<b> such that </b><br/><a NAME="E2:25_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</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="rsspace.html#K10">Zero_</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>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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>,<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> #)
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">s</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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> <font color="Maroon">f</font>,<font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="rsspace.html#K10">Zero_</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>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
;<br/>


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

<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font>, <font color="Maroon">yy</font> = <font color="Maroon">y</font>, <font color="Maroon">zz</font> = <font color="Maroon">z</font> as    <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>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">yy</font></span>)</span>,<font color="Maroon">zz</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">yy</font></span>)</span> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">zz</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">xx</font> <a href="lopban_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">zz</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">xx</font>,<span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">zz</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="lopban_2.html#K7" target="_self">FuncUnit</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">xx</font> <a href="lopban_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="lopban_2.html#K7" target="_self">FuncUnit</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="lopban_2.html#K8" target="_self">Ring_of_BoundedLinearOperators</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="lopban_2.html#K7" target="_self">FuncUnit</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_2.html#K7" target="_self">FuncUnit</a> <font color="Maroon">X</font></span>)</span> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">xx</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p3">(<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="p3">(<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> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">xx</font> <a href="lopban_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="lopban_2.html#K3" target="_self">+</a> <font color="Maroon">zz</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">yy</font></span>)</span> <a href="lopban_2.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">zz</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">yy</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">xx</font>,<font color="Maroon">zz</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p3">(<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="p3">(<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> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="lopban_2.html#K3" target="_self">+</a> <font color="Maroon">zz</font></span>)</span> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">xx</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">xx</font></span>)</span> <a href="lopban_2.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">zz</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">xx</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="lopban_2.html#K4" target="_self">*</a> <font color="Maroon">xx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">zz</font>,<font color="Maroon">xx</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</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">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_2.html#K6" target="_self">FuncMult</a> <font color="Maroon">X</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>


</div>
