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<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">X0</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>reconsider </b><font color="Maroon">A0</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</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/>
<b>reconsider </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</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/>
<b>reconsider </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</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/>
<b>thus </b><a NAME="E4:42"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> iff ( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_1"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> implies ( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_1"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font>
 ;<br/>

<a NAME="E2:42_1_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#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A0</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>
<a NAME="E3:42_1_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#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</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="E4:42_1_1"/><b>then </b>
<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> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#D2">TSEP_1:def 2</a></i>;<br/>
<a NAME="E5:42_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E6:42_1_1"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</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="E7:42_1_1"/><b>then </b>
( <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A0</font> &amp; <font color="Maroon">A2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A0</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="E8:42_1_1"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>


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

<b>thus </b><a NAME="E2:42_1"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> implies <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_2"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> &amp; <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font> )
 ;<br/>

<a NAME="E2:42_1_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A0</font> &amp; <font color="Maroon">A2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A0</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>
<a NAME="E3:42_1_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</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="E4:42_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E5:42_1_2"/><b>then </b>
<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> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E6:42_1_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#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#D2">TSEP_1:def 2</a></i>;<br/>
<a NAME="E7:42_1_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#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A0</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="E8:42_1_2"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X0</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>


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


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

<b>hence </b><a NAME="E5:42"/>
( <font color="Maroon">X0</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> iff ( <font color="Maroon">X0</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X0</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font> ) )
 ;<br/>


</div>
