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

<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>, <font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>, <font color="Maroon">Y</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/>

<b>assume </b><a NAME="E1:22"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">V</font> is    <a href="rusub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> is    <a href="rusub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">Y</font> )
 ;<br/>

<b>thus </b><a NAME="E2:22"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rusub_1.html#D1" target="_self">RUSUB_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<a NAME="E1:22_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:22"><i><font color="Green">E19</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E2:22_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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

<b>thus </b><a NAME="E3:22"/>
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<a NAME="E1:22_2"/>
( the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">X</font> &amp; the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:22"><i><font color="Green">E19</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E2:22_2"/>
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">Y</font>
 ;<br/>


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

<b>thus </b><a NAME="E4:22"/>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">AV</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>;<br/>
<b>set </b><font color="Maroon">VV</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>
<b>set </b><font color="Maroon">AX</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">VX</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">AY</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font>;<br/>
<a NAME="E1:22_3"/>
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:22"><i><font color="Green">E19</font></i></a>, </i>;<br/>
<a NAME="E2:22_3"/><b>then </b>
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E3:22_3"/>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>


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

<b>thus </b><a NAME="E5:22"/>
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">Y</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">MV</font> = the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>;<br/>
<b>set </b><font color="Maroon">VV</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>
<b>set </b><font color="Maroon">MX</font> = the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">VX</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">MY</font> = the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">Y</font>;<br/>
<a NAME="E1:22_4"/>
( the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">X</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">Y</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:22"><i><font color="Green">E19</font></i></a>, </i>;<br/>
<a NAME="E2:22_4"/><b>then </b>
( the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">Y</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<b>hence </b><a NAME="E3:22_4"/>
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">Y</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>


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

<b>set </b><font color="Maroon">SV</font> = the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font>;<br/>
<b>set </b><font color="Maroon">VV</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>
<b>set </b><font color="Maroon">SX</font> = the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">VX</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">SY</font> = the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">Y</font>;<br/>
<a NAME="E6:22"/>
( the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">Y</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:22"><i><font color="Green">E19</font></i></a>, </i>;<br/>
<a NAME="E7:22"/><b>then </b>
( the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">Y</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E8:22"/>
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">Y</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>


</div>
