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

<b>let </b><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green" title="E29">A10</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> iff <font color="Olive" title="b1">x</font> is  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> )
 ;<br/>

<b>assume </b><a NAME="E2:33_1_2"/><i><font color="Green" title="E30">A11</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> iff <font color="Olive" title="b1">x</font> is  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:33_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font>
 ;<br/>

<a NAME="E2:33_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> is  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:33_1_2"><i><font color="Green" title="E29">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:33_1_2"><i><font color="Green" title="E30">A11</font></i></a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:33_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font>
 ;<br/><a NAME="E3:33_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> is  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:33_1_2"><i><font color="Green" title="E30">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E4:33_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:33_1_2"><i><font color="Green" title="E29">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:33_1_2"/>
<font color="Maroon" title="c2">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div>
