<?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">Y</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>assume </b><a NAME="E1:45"/><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/>

<a NAME="E2:45"/>
( <font color="Maroon">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:45"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> 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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:45"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> 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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> <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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</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#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E5:45"/><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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> <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#K1" target="_self">union</a> <font color="Maroon">Y</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#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E6:45"/><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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> <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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E7:45"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2" target="_self">Th2</a></i>;<br/>
<b>thus </b><a NAME="E8:45"/>
<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#K1" target="_self">union</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> <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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

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#K1" target="_self">union</a> <font color="Maroon">Y</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#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</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#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
<b>by </b><i>, </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#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <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#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T24">XBOOLE_1:24</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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> <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#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> <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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> <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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E2:45_1"/>
<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#K1" target="_self">union</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span> <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#K1" target="_self">union</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>hence </b><a NAME="E9:45"/>
<font color="Maroon">Y</font> <a href="tsep_1.html#K1" target="_self">union</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X1</font></span>)</span> <a href="tsep_1.html#K2" target="_self">meet</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font></span>)</span>
 ;<br/>


</div>
