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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">X2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">X3</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>







<b>thus </b><a NAME="E1:38"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> implies <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font>
 ;<br/>

<b>then </b>the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
<b>by </b><i/>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X1</font></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T4" target="_self">Th4</a>, </i>
;<br/>
<b>hence </b><a NAME="E3:38_1"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X1</font>
 <b>by </b><i/>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E27</font></i>: 
( ( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> ) or ( <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font> ) )
 ;<br/><div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> )
 ;<br/><a NAME="E2:38_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E3:38_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:38_2_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E5:38_2_1"/><b>then </b><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/><a NAME="E6:38_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E7:38_2_1"/><b>then </b><i><font color="Green">E32</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E8:38_2_1"/><b>then </b>
<font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E9:38_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:38_2"><i><font color="Green">E27</font></i></a>, </i>;<br/><a NAME="E10:38_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E11:38_2_1"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font> )
 <b>by </b><i>, , <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font> )
 ;<br/><a NAME="E2:38_2_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E3:38_2_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:38_2_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:38_2"><i><font color="Green">E27</font></i></a>, </i>;<br/><a NAME="E5:38_2_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/><a NAME="E6:38_2_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E7:38_2_2"/><b>then </b><i><font color="Green">E66</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E8:38_2_2"/><b>then </b>
<font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E9:38_2_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E10:38_2_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E11:38_2_2"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:38_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tsep_1.html#E2:38_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><b>then </b>the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font>
<b>by </b><i><a class="ref" href="tsep_1.html#T4" target="_self">Th4</a>, </i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font>
<b>by </b><i><a class="ref" href="tsep_1.html#T4" target="_self">Th4</a>, , </i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X3</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T4" target="_self">Th4</a>, , </i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T4" target="_self">Th4</a>, , </i>
;<br/><b>hence </b><a NAME="E5:38_2"/>
<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
( ( ( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> ) or ( <font color="Maroon">X2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font> ) ) implies <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">X3</font></span>)</span> )
 ;<br/>


</div>
