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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="bhsp_1.html#V1">strict</a>  <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>thus </b><a NAME="E1:20"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:20_1"/><b>then </b><i><font color="Green">E20</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/>
<a NAME="E3:20_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rusub_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:20_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E2:20_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E5:20_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rusub_1.html#T24">RUSUB_1:24</a></i>;<br/>


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

<b>thus </b><a NAME="E2:20"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>3</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="rusub_2.html#T16" target="_self">Th16</a></i>;<br/>


</div>
