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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">V1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">d1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">D</font>;<br/><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">D</font>;<br/><b>let </b><font color="Maroon" title="c6">M</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c3">D</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c4">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">V1</font>
 <b>and </b><br/><a NAME="E4:26"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">W</font> =  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>
<a NAME="E5:26"/><i><font color="Green" title="E23">A5</font></i>: 
 <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>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E2:26"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>
<a NAME="E6:26"/><i><font color="Green" title="E24">A6</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>

<b>thus </b><font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">y</font> = 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="rlsub_1.html#E3:26"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>

;<br/>


</div><b>end;</b></div>
<a NAME="E7:26"/><i><font color="Green" title="E25">A7</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c9">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>



<b>thus </b><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c9">x</font> = 
the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="rlsub_1.html#E4:26"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">x</font>

;<br/>


</div><b>end;</b></div>
<a NAME="E8:26"/>
(  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> &amp;  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp;  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp;  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> &amp;  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">RealLinearSpace-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c8">AV</font> = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c9">MV</font> = the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>thus </b><a NAME="E1:26_3"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b1">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c11">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c11">y</font> = 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c13">y1</font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">x1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c11">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E2:26_3"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c12">z</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c14">y1</font> = <font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c15">z1</font> = <font color="Maroon" title="c12">z</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c12">z</font> = 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c11">y</font></span>)</span>,<font color="Maroon" title="c15">z1</font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">z1</font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">z1</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/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c12">z</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c12">z</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E3:26_3"/>
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <font color="Olive" title="b1">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> <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c12">z</font> =  <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>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c10">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> <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #)</span>)</span> = 
<font color="Maroon" title="c11">y</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> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E2:26"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>


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

<b>thus </b><a NAME="E4:26_3"/>
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #)  ex <font color="Olive" title="b2">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) st <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">y</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>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #)
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c10">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:26_3_4"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c11">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></i>;<br/>
<font color="Maroon" title="c12">v</font> = 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c11">x1</font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E3:26_3_4"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c11">x1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) ;<br/>
<b>take </b>
<font color="Maroon" title="c13">y</font>
;<br/>

<b>thus </b><a NAME="E6:26_3_4"/>
<font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c13">y</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>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #)
 <b>by </b><i><a class="txt" href="rlsub_1.html#E2:26"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="rlsub_1.html#E3:26_3_4"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E5:26_3"/>
for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">y</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>



<b>reconsider </b><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c14">y1</font> = <font color="Maroon" title="c12">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c12">y</font></span>)</span> = 
the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">y1</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c13">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c13">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c11">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c11">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="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E6:26_3"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c12">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>





<b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c13">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c13">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">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="c11">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:26"><i><font color="Green" title="E24">A6</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E7:26_3"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c12">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>





<b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c13">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
;<br/>


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

<b>let </b><font color="Maroon" title="c10">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b>1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c10">x</font> = 
1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c11">y</font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:26"><i><font color="Green" title="E25">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:26"/>
 <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rlsub_1.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="rlsub_1.html#E3:26"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="rlsub_1.html#E4:26"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="rlsub_1.html#E5:26"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="rlsub_1.html#D2" target="_self" title="RLSUB_1:def.2">Def2</a></i>;<br/>


</div>
