<?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">W1</font>, <font color="Maroon" title="c3">W2</font> be    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>set </b><font color="Maroon" title="c4">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> ) </span>}</span> ;<br/>
<a NAME="E1:6"/><i><font color="Green" title="E7">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> ) </span>}</span>  &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="rusub_2.html#D1" target="_self" title="RUSUB_2:def.1">Def1</a></i>;<br/>
<a NAME="E2:6"/><i><font color="Green" title="E8">A2</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">u</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="E3:6_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">u</font>
 <b>and </b><br/><a NAME="E4:6_1"/><i><font color="Green" title="E10">A4</font></i>: 
( <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c8">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> )
 ;<br/>
<b>thus </b><a NAME="E5:6_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rusub_2.html#E3:6_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="rusub_2.html#E4:6_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">u</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="E3:6_2"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">u</font>
 <b>and </b><br/><a NAME="E4:6_2"/><i><font color="Green" title="E10">A6</font></i>: 
( <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> &amp; <font color="Maroon" title="c8">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> )
 ;<br/>
<b>thus </b><a NAME="E5:6_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rusub_2.html#E3:6_2"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="rusub_2.html#E4:6_2"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rusub_2.html#E2:6"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:6"/>
<font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c2">W1</font>
 <b>by </b><i><a class="txt" href="rusub_2.html#E1:6"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="rusub_1.html#T24" title="RUSUB_1:th.24">RUSUB_1:24</a></i>;<br/>


</div>
