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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">Y1</font>, <font color="Maroon" title="c4">X2</font>, <font color="Maroon" title="c5">Y2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>









<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">Y1</font> &amp; <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c5">Y2</font> )
 ;<br/>

<b>assume </b><a NAME="E2:33"/><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c3">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c5">Y2</font> or <font color="Maroon" title="c3">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E27">A3</font></i>: 
not <font color="Maroon" title="c3">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c5">Y2</font>
 ;<br/><div><i><font color="Green" title="E28">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1"/>
<font color="Maroon" title="c3">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c4">X2</font>
 ;<br/><a NAME="E2:33_1_1"/><b>then </b>
( <font color="Maroon" title="c3">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="tmap_1.html#T34" target="_self" title="TMAP_1:th.34">Th34</a>, <a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:33"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="tmap_1.html#E1:33_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="tmap_1.html#T23" target="_self" title="TMAP_1:th.23">Th23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_2"/>
<font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font>
 ;<br/><a NAME="E2:33_1_2"/><b>then </b>
( <font color="Maroon" title="c3">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">X1</font> &amp; <font color="Maroon" title="c2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="tmap_1.html#T34" target="_self" title="TMAP_1:th.34">Th34</a>, <a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></i>;<br/><b>hence </b><a NAME="E3:33_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:33"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="tmap_1.html#E1:33_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="tmap_1.html#T23" target="_self" title="TMAP_1:th.23">Th23</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1"/>
( <font color="Maroon" title="c3">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c2">X1</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:33_1"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
( <font color="Maroon" title="c3">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c2">X1</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="tmap_1.html#T24" target="_self" title="TMAP_1:th.24">Th24</a></i>;<br/>


</div>
