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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</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>let </b><font color="Maroon" title="c7">S</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"><font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c3">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green" title="E15">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="E2:19"/><i><font color="Green" title="E16">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="E3:19"/><i><font color="Green" title="E17">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="E4:19"/><i><font color="Green" title="E18">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>
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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>
 ;<br/>

<a NAME="E6:19"/>
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c8">W</font> =  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #);<br/>
<a NAME="E1:19_1"/><i><font color="Green" title="E20">A6</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E2:19_1"/><i><font color="Green" title="E21">A7</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</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="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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="rusub_1.html#E3:19"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E3:19_1"/><i><font color="Green" title="E22">A8</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</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="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="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="rusub_1.html#E4:19"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E4:19_1"/><i><font color="Green" title="E23">A9</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) holds  <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #);<br/>

<a NAME="E1:19_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #)</span>)</span> <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="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="rusub_1.html#E5:19"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:19_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_1"/>
(  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="bhsp_1.html#V2" title="BHSP_1:attr.2">RealUnitarySpace-like</a> &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">RealLinearSpace-like</a> &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c9">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="c10">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>set </b><font color="Maroon" title="c11">SV</font> = the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font>;<br/>
<a NAME="E1:19_1_2"/><i><font color="Green" title="E24">A10</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c15">y1</font> = <font color="Maroon" title="c13">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="rusub_1.html#E1:19"><i><font color="Green" title="E15">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="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c13">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="c14">x1</font>,<font color="Maroon" title="c15">y1</font>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green" title="E21">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">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="c13">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green" title="E21">A7</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:19_1_2"/><i><font color="Green" title="E25">A11</font></i>: 
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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font>, <font color="Maroon" title="c14">z</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c13">y</font>, <font color="Maroon" title="c17">z1</font> = <font color="Maroon" title="c14">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="rusub_1.html#E1:19"><i><font color="Green" title="E15">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="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c14">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="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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c13">y</font></span>)</span>,<font color="Maroon" title="c17">z1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green" title="E21">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">z1</font>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green" title="E21">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">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="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="c15">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c14">z</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green" title="E21">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c14">z</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green" title="E21">A7</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:19_1_2"/><i><font color="Green" title="E26">A12</font></i>: 
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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_3"><b>proof </b></a><div class="add">

<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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c14">z</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</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="rusub_1.html#E1:19"><i><font color="Green" title="E15">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="c12">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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #)</span>)</span> = 
<font color="Maroon" title="c13">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="rusub_1.html#E2:19_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="rusub_1.html#E2:19"><i><font color="Green" title="E16">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">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>
<a NAME="E4:19_1_2"/><i><font color="Green" title="E27">A13</font></i>: 
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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_4"><b>proof </b></a><div class="add">

<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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c13">x1</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="rusub_1.html#E1:19"><i><font color="Green" title="E15">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="c14">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="E3:19_1_2_4"/><i><font color="Green" title="E28">A14</font></i>: 
<font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">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="c14">v</font> = 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c13">x1</font>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1_2_4"><i><font color="Green" title="E28">A14</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="c13">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="c12">x</font>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green" title="E22">A8</font></i></a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) ;<br/>
<b>take </b>
<font color="Maroon" title="c15">y</font>
;<br/>

<b>thus </b><a NAME="E6:19_1_2_4"/>
<font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c15">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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #)
 <b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="rusub_1.html#E2:19"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="rusub_1.html#E3:19_1_2_4"><i><font color="Green" title="E28">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_1_2"/><i><font color="Green" title="E28">A15</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>, <font color="Olive" title="b3">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #);<br/>



<b>reconsider </b><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c14">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="rusub_1.html#E1:19"><i><font color="Green" title="E15">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="c12">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">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c14">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="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="c12">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c14">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="rusub_1.html#E4:19"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">y1</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green" title="E21">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c15">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="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c16">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="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"><span class="p2">(<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c15">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green" title="E22">A8</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="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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c13">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="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">y</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="rusub_1.html#E3:19"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:19_1_2"/><i><font color="Green" title="E29">A16</font></i>: 
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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="19_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c14">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #);<br/>



<b>reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">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="rusub_1.html#E1:19"><i><font color="Green" title="E15">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="c12">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c15">y</font>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c15">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="c13">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c15">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="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"><span class="p2">(<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c15">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green" title="E22">A8</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="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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">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="c13">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="rusub_1.html#E3:19"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:19_1_2"/><i><font color="Green" title="E30">A17</font></i>: 
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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="19_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c14">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #);<br/>



<b>reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">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="rusub_1.html#E1:19"><i><font color="Green" title="E15">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="c12">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c15">y</font>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">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">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c15">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="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="c12">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">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">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="rusub_1.html#E4:19"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:19_1_2"/><i><font color="Green" title="E31">A18</font></i>: 
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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) holds  1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_8"><b>proof </b></a><div class="add">

<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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</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="rusub_1.html#E1:19"><i><font color="Green" title="E15">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="c12">x</font> = 
1 <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="rusub_1.html#E3:19_1"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">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>
<a NAME="E9:19_1_2"/>
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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #)<br/> for <font color="Olive" title="b4">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies <font color="Olive" title="b1">x</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) ) &amp; ( <font color="Olive" title="b1">x</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) implies <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_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="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b1">x</font> &amp; <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="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_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="b4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font>, <font color="Maroon" title="c14">z</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #);<br/><b>let </b><font color="Maroon" title="c15">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>reconsider </b><font color="Maroon" title="c16">x1</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="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">y1</font> = <font color="Maroon" title="c13">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="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">z1</font> = <font color="Maroon" title="c14">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="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:19_1_2_9"/><i><font color="Green" title="E32">A19</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies <font color="Maroon" title="c12">x</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_2_9_1"/>
<font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:19_1_2_9_1"/><b>then </b>
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c16">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>
<a NAME="E3:19_1_2_9_1"/><b>then </b>
<font color="Maroon" title="c16">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1_2_9_1"/>
<font color="Maroon" title="c12">x</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #)
 <b>by </b><i><a class="txt" href="rusub_1.html#E2:19"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_1_2_9"/>
( <font color="Maroon" title="c12">x</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) implies <font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_2_9_2"/>
<font color="Maroon" title="c12">x</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #)
 ;<br/>

<a NAME="E2:19_1_2_9_2"/><b>then </b>
<font color="Maroon" title="c16">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>;<br/>
<a NAME="E3:19_1_2_9_2"/><b>then </b>
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c16">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1_2_9_2"/>
<font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:19_1_2_9"/>
( <font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 iff <font color="Maroon" title="c12">x</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="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) )
 <b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1_2_9"><i><font color="Green" title="E32">A19</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:19_1_2_9"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c12">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_3"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_9_3"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c16">x1</font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>;<br/>
<a NAME="E2:19_1_2_9_3"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c16">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_2_9_3"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E8:19_1_2_9"/>
<font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c12">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_4"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_9_4"/>
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c17">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c16">x1</font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>;<br/>
<a NAME="E2:19_1_2_9_4"/><b>then </b>
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c17">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">y1</font>,<font color="Maroon" title="c16">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>;<br/>
<a NAME="E3:19_1_2_9_4"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">y1</font>,<font color="Maroon" title="c16">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:19_1_2_9_4"/>
<font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E9:19_1_2_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">z</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">z</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_5"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_9_5"/><i><font color="Green" title="E33">A20</font></i>: 
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">y1</font>,<font color="Maroon" title="c18">z1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c18">z1</font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>;<br/>
<i><font color="Green" title="E34">A21</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">z</font> = 
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c13">y</font></span>)</span>,<font color="Maroon" title="c14">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green" title="E23">A9</font></i></a></i>
<br/>.= 
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">y1</font></span>)</span>,<font color="Maroon" title="c14">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green" title="E21">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c18">z1</font>
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c18">z1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c18">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">z</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">z</font></span>)</span> = 
<span class="p1">(<span class="default">the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">z</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green" title="E23">A9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green" title="E23">A9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c18">z1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c18">z1</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19_1_2_9_5"><i><font color="Green" title="E33">A20</font></i></a>, <a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:19_1_2_9_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">z</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c14">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1_2_9_5"><i><font color="Green" title="E34">A21</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E10:19_1_2_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">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="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_6"><b>proof </b></a><div class="add">

<i><font color="Green" title="E33">A22</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">y</font> = 
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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"><span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">x</font></span>)</span>,<font color="Maroon" title="c13">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green" title="E23">A9</font></i></a></i>
<br/>.= 
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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"><span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c16">x1</font></span>)</span>,<font color="Maroon" title="c13">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c16">x1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c17">y1</font>
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c15">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c17">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>
;<br/>
<font color="Maroon" title="c15">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">y</font></span>)</span> = 
<font color="Maroon" title="c15">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green" title="E23">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c17">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:19_1_2_9_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">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="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c13">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="rusub_1.html#E1:19_1_2_9_6"><i><font color="Green" title="E33">A22</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:19_1_2"/>
(  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="bhsp_1.html#V2" title="BHSP_1:attr.2">RealUnitarySpace-like</a> &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">RealLinearSpace-like</a> &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp;  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> )
 <b>by </b><i><a class="txt" href="rusub_1.html#E1:19_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="rusub_1.html#E2:19_1_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="rusub_1.html#E3:19_1_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="rusub_1.html#E4:19_1_2"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="rusub_1.html#E5:19_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="rusub_1.html#E6:19_1_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="rusub_1.html#E7:19_1_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="txt" href="rusub_1.html#E8:19_1_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a>, <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="E6:19_1"/>
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="rusub_1.html#E3:19"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="rusub_1.html#E4:19"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="rusub_1.html#E5:19"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="rusub_1.html#E1:19_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="rusub_1.html#D1" target="_self" title="RUSUB_1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:19"/>
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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>,<font color="Maroon" title="c7">S</font> #) is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 ;<br/>


</div>
