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<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">Y1</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">Y2</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="E1:38"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">X1</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> )
 ;<br/>

<b>assume </b><a NAME="E2:38"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font>
 ;<br/>

<b>assume </b><a NAME="E3:38"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>, <font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>, <font color="Maroon">C1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y1</font>, <font color="Maroon">C2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E5:38"/><i><font color="Green">E36</font></i>: 
( 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">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#D2">TSEP_1:def 2</a></i>;<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">Y1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 ;<br/><a NAME="E2:38_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">C1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/><a NAME="E3:38_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><b>then </b><i><font color="Green">E42</font></i>: <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tmap_1.html#E3:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
<font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>

;<br/><a NAME="E5:38_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</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#T15">XBOOLE_1:15</a></i>;<br/><a NAME="E6:38_1"/><b>then </b>
<font color="Maroon">C2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E5:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E7:38_1"/><b>then </b>
<font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/><a NAME="E8:38_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#D5">TSEP_1:def 5</a></i>;<br/><a NAME="E9:38_1"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#D1" target="_self">Def1</a>, <a class="txt" href="tmap_1.html#E5:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E10:38_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E11:38_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></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"/>
<font color="Maroon">Y2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 ;<br/><a NAME="E2:38_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">C2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/><a NAME="E3:38_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><b>then </b><i><font color="Green">E44</font></i>: <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="tmap_1.html#E3:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>

;<br/><a NAME="E5:38_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</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#T15">XBOOLE_1:15</a></i>;<br/><a NAME="E6:38_2"/><b>then </b>
<font color="Maroon">C1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E5:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E7:38_2"/><b>then </b>
<font color="Maroon">Y1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/><a NAME="E8:38_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#D5">TSEP_1:def 5</a></i>;<br/><a NAME="E9:38_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#D1" target="_self">Def1</a>, <a class="txt" href="tmap_1.html#E5:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E10:38_2"/><b>then </b>
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E11:38_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:38"/>
( <font color="Maroon">Y1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> )
 <b>by </b><i/>;<br/>


</div>
