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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:36"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:36_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:36"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a></i>;<br/><b>thus </b><a NAME="E3:36_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:36_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:36"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/>
<a NAME="E2:36_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E3:36_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:36"><i><font color="Green">E28</font></i></a>, <a class="txt" href="tmap_1.html#E1:36_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tmap_1.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E4:36_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:36_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:36_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:36_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tsep_1.html#T29">TSEP_1:29</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:36_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:36"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a></i>;<br/><b>thus </b><a NAME="E3:36_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:36_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:36"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/>
<a NAME="E2:36_2_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E3:36_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:36"><i><font color="Green">E28</font></i></a>, <a class="txt" href="tmap_1.html#E1:36_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tmap_1.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E4:36_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:36_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:36_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:36_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tsep_1.html#T29">TSEP_1:29</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:36"/>
( ( <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font> implies ( <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font> ) ) &amp; ( <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font> implies ( <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:36"><i><font color="Green">E29</font></i></a></i>;<br/>


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