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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">IT</font> =  <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/>
<a NAME="E1:35"/>
(  <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) is <a href="rlvect_1.html#V6">right_complementable</a> &amp;  <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) is <a href="rlvect_1.html#V7">RealLinearSpace-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_1"/>
for <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) holds  <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">u</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/>



<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a>  ;<br/>
<b>thus </b><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">u'</font>,<font color="Maroon">v'</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">v'</font>,<font color="Maroon">u'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font>

;<br/>


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

<b>thus </b><a NAME="E2:35_1"/>
for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/><b>let </b><font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/>





<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font>, <font color="Maroon">w'</font> = <font color="Maroon">w</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a>  ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">u'</font>,<font color="Maroon">v'</font></span>)</span>,<font color="Maroon">w'</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">v'</font>,<font color="Maroon">w'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>

;<br/>


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

<b>thus </b><a NAME="E3:35_1"/>
for <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) holds  <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</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="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">u</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a>  ;<br/>
<b>thus </b><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</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="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #)</span>)</span> = 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">u'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E4:35_1"/>
for <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #)  ex <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) st <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</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="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #)
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<font color="Maroon">u'</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) ;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<font color="Maroon">u'</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>

<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #)
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E5:35_1"/>
for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">u</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/>





<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">u'</font></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">w'</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">v'</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) ;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">u'</font>,<font color="Maroon">v'</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">w</font>,<font color="Maroon">w'</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>

;<br/>


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

<b>thus </b><a NAME="E6:35_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K3">+</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #);<br/>





<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">v'</font></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">w'</font> = <span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">v'</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">w</font>,<font color="Maroon">w'</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>

;<br/>


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

<b>thus </b><a NAME="E7:35_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E8:35_1"/>
for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) holds  1 <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:35"/>
 <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> #) is   <a href="rlvect_1.html#NM2">RealLinearSpace</a>
 ;<br/>


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